Integrand size = 252, antiderivative size = 33 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{\frac {1}{2} \left (\frac {6}{5-16 \left (\frac {2-x}{x}-x\right )^4}-x\right )} \] Output:
exp(3/(5-4*((2-x)/x-x)^2*(2*(2-x)/x-2*x)^2)-1/2*x)
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{-\frac {x}{2}-\frac {3 x^4}{256-512 x-128 x^2+640 x^3+11 x^4-320 x^5-32 x^6+64 x^7+16 x^8}} \] Input:
Integrate[(E^((-256*x + 512*x^2 + 128*x^3 - 646*x^4 - 11*x^5 + 320*x^6 + 3 2*x^7 - 64*x^8 - 16*x^9)/(512 - 1024*x - 256*x^2 + 1280*x^3 + 22*x^4 - 640 *x^5 - 64*x^6 + 128*x^7 + 32*x^8))*(-65536 + 262144*x - 196608*x^2 - 46489 6*x^3 + 642560*x^4 + 340480*x^5 - 721920*x^6 - 161536*x^7 + 456711*x^8 + 8 0384*x^9 - 178368*x^10 - 41984*x^11 + 39584*x^12 + 14336*x^13 - 3072*x^14 - 2048*x^15 - 256*x^16))/(131072 - 524288*x + 393216*x^2 + 917504*x^3 - 12 66688*x^4 - 677888*x^5 + 1436160*x^6 + 323072*x^7 - 917262*x^8 - 161536*x^ 9 + 359040*x^10 + 84736*x^11 - 79168*x^12 - 28672*x^13 + 6144*x^14 + 4096* x^15 + 512*x^16),x]
Output:
E^(-1/2*x - (3*x^4)/(256 - 512*x - 128*x^2 + 640*x^3 + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8))
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.
| \(\displaystyle \int \frac {\left (-256 x^{16}-2048 x^{15}-3072 x^{14}+14336 x^{13}+39584 x^{12}-41984 x^{11}-178368 x^{10}+80384 x^9+456711 x^8-161536 x^7-721920 x^6+340480 x^5+642560 x^4-464896 x^3-196608 x^2+262144 x-65536\right ) \exp \left (\frac {-16 x^9-64 x^8+32 x^7+320 x^6-11 x^5-646 x^4+128 x^3+512 x^2-256 x}{32 x^8+128 x^7-64 x^6-640 x^5+22 x^4+1280 x^3-256 x^2-1024 x+512}\right )}{512 x^{16}+4096 x^{15}+6144 x^{14}-28672 x^{13}-79168 x^{12}+84736 x^{11}+359040 x^{10}-161536 x^9-917262 x^8+323072 x^7+1436160 x^6-677888 x^5-1266688 x^4+917504 x^3+393216 x^2-524288 x+131072} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {\left (-256 x^{16}-2048 x^{15}-3072 x^{14}+14336 x^{13}+39584 x^{12}-41984 x^{11}-178368 x^{10}+80384 x^9+456711 x^8-161536 x^7-721920 x^6+340480 x^5+642560 x^4-464896 x^3-196608 x^2+262144 x-65536\right ) \exp \left (\frac {-16 x^9-64 x^8+32 x^7+320 x^6-11 x^5-646 x^4+128 x^3+512 x^2-256 x}{32 x^8+128 x^7-64 x^6-640 x^5+22 x^4+1280 x^3-256 x^2-1024 x+512}\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {\exp \left (-\frac {16 x^9+64 x^8-32 x^7-320 x^6+11 x^5+646 x^4-128 x^3-512 x^2+256 x}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right ) \left (256 x^{16}+2048 x^{15}+3072 x^{14}-14336 x^{13}-39584 x^{12}+41984 x^{11}+178368 x^{10}-80384 x^9-456711 x^8+161536 x^7+721920 x^6-340480 x^5-642560 x^4+464896 x^3+196608 x^2-262144 x+65536\right )}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {\exp \left (-\frac {16 x^9+64 x^8-32 x^7-320 x^6+11 x^5+646 x^4-128 x^3-512 x^2+256 x}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right ) \left (256 x^{16}+2048 x^{15}+3072 x^{14}-14336 x^{13}-39584 x^{12}+41984 x^{11}+178368 x^{10}-80384 x^9-456711 x^8+161536 x^7+721920 x^6-340480 x^5-642560 x^4+464896 x^3+196608 x^2-262144 x+65536\right )}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{2} \int \frac {\exp \left (-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right ) \left (256 x^{16}+2048 x^{15}+3072 x^{14}-14336 x^{13}-39584 x^{12}+41984 x^{11}+178368 x^{10}-80384 x^9-456711 x^8+161536 x^7+721920 x^6-340480 x^5-642560 x^4+464896 x^3+196608 x^2-262144 x+65536\right )}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (-\frac {24 \exp \left (-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right ) \left (x^3-x^2+5 x-7\right )}{16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256}+\exp \left (-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right )-\frac {24 \exp \left (-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}\right ) \left (277 x^7+587 x^6-1463 x^5-2355 x^4+4096 x^3+1920 x^2-4864 x+1792\right )}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\int e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}}dx+43008 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}}}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx-116736 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx+46080 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^2}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx+98304 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^3}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx-56520 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^4}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx-35112 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^5}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx+14088 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^6}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx+6648 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^7}{\left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )^2}dx-168 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}}}{16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256}dx+120 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x}{16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256}dx-24 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^2}{16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256}dx+24 \int \frac {e^{-\frac {x \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+646 x^3-128 x^2-512 x+256\right )}{2 \left (16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256\right )}} x^3}{16 x^8+64 x^7-32 x^6-320 x^5+11 x^4+640 x^3-128 x^2-512 x+256}dx\right )\) |
Input:
Int[(E^((-256*x + 512*x^2 + 128*x^3 - 646*x^4 - 11*x^5 + 320*x^6 + 32*x^7 - 64*x^8 - 16*x^9)/(512 - 1024*x - 256*x^2 + 1280*x^3 + 22*x^4 - 640*x^5 - 64*x^6 + 128*x^7 + 32*x^8))*(-65536 + 262144*x - 196608*x^2 - 464896*x^3 + 642560*x^4 + 340480*x^5 - 721920*x^6 - 161536*x^7 + 456711*x^8 + 80384*x ^9 - 178368*x^10 - 41984*x^11 + 39584*x^12 + 14336*x^13 - 3072*x^14 - 2048 *x^15 - 256*x^16))/(131072 - 524288*x + 393216*x^2 + 917504*x^3 - 1266688* x^4 - 677888*x^5 + 1436160*x^6 + 323072*x^7 - 917262*x^8 - 161536*x^9 + 35 9040*x^10 + 84736*x^11 - 79168*x^12 - 28672*x^13 + 6144*x^14 + 4096*x^15 + 512*x^16),x]
Output:
$Aborted
Time = 3.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.64
| method | result | size |
| gosper | \({\mathrm e}^{-\frac {x \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+646 x^{3}-128 x^{2}-512 x +256\right )}{2 \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256\right )}}\) | \(87\) |
| risch | \({\mathrm e}^{-\frac {x \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+646 x^{3}-128 x^{2}-512 x +256\right )}{2 \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256\right )}}\) | \(87\) |
| parallelrisch | \({\mathrm e}^{-\frac {x \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+646 x^{3}-128 x^{2}-512 x +256\right )}{2 \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256\right )}}\) | \(87\) |
| orering | \(-\frac {2 \left (16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256\right )^{2} \left (-256 x^{16}-2048 x^{15}-3072 x^{14}+14336 x^{13}+39584 x^{12}-41984 x^{11}-178368 x^{10}+80384 x^{9}+456711 x^{8}-161536 x^{7}-721920 x^{6}+340480 x^{5}+642560 x^{4}-464896 x^{3}-196608 x^{2}+262144 x -65536\right ) {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}}{\left (256 x^{16}+2048 x^{15}+3072 x^{14}-14336 x^{13}-39584 x^{12}+41984 x^{11}+178368 x^{10}-80384 x^{9}-456711 x^{8}+161536 x^{7}+721920 x^{6}-340480 x^{5}-642560 x^{4}+464896 x^{3}+196608 x^{2}-262144 x +65536\right ) \left (512 x^{16}+4096 x^{15}+6144 x^{14}-28672 x^{13}-79168 x^{12}+84736 x^{11}+359040 x^{10}-161536 x^{9}-917262 x^{8}+323072 x^{7}+1436160 x^{6}-677888 x^{5}-1266688 x^{4}+917504 x^{3}+393216 x^{2}-524288 x +131072\right )}\) | \(377\) |
| norman | \(\frac {-512 x \,{\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}-128 x^{2} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}+640 x^{3} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}+11 x^{4} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}-320 x^{5} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}-32 x^{6} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}+64 x^{7} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}+16 x^{8} {\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}+256 \,{\mathrm e}^{\frac {-16 x^{9}-64 x^{8}+32 x^{7}+320 x^{6}-11 x^{5}-646 x^{4}+128 x^{3}+512 x^{2}-256 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}}}{16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256}\) | \(877\) |
Input:
int((-256*x^16-2048*x^15-3072*x^14+14336*x^13+39584*x^12-41984*x^11-178368 *x^10+80384*x^9+456711*x^8-161536*x^7-721920*x^6+340480*x^5+642560*x^4-464 896*x^3-196608*x^2+262144*x-65536)*exp((-16*x^9-64*x^8+32*x^7+320*x^6-11*x ^5-646*x^4+128*x^3+512*x^2-256*x)/(32*x^8+128*x^7-64*x^6-640*x^5+22*x^4+12 80*x^3-256*x^2-1024*x+512))/(512*x^16+4096*x^15+6144*x^14-28672*x^13-79168 *x^12+84736*x^11+359040*x^10-161536*x^9-917262*x^8+323072*x^7+1436160*x^6- 677888*x^5-1266688*x^4+917504*x^3+393216*x^2-524288*x+131072),x,method=_RE TURNVERBOSE)
Output:
exp(-1/2*x*(16*x^8+64*x^7-32*x^6-320*x^5+11*x^4+646*x^3-128*x^2-512*x+256) /(16*x^8+64*x^7-32*x^6-320*x^5+11*x^4+640*x^3-128*x^2-512*x+256))
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.70 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{\left (-\frac {16 \, x^{9} + 64 \, x^{8} - 32 \, x^{7} - 320 \, x^{6} + 11 \, x^{5} + 646 \, x^{4} - 128 \, x^{3} - 512 \, x^{2} + 256 \, x}{2 \, {\left (16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256\right )}}\right )} \] Input:
integrate((-256*x^16-2048*x^15-3072*x^14+14336*x^13+39584*x^12-41984*x^11- 178368*x^10+80384*x^9+456711*x^8-161536*x^7-721920*x^6+340480*x^5+642560*x ^4-464896*x^3-196608*x^2+262144*x-65536)*exp((-16*x^9-64*x^8+32*x^7+320*x^ 6-11*x^5-646*x^4+128*x^3+512*x^2-256*x)/(32*x^8+128*x^7-64*x^6-640*x^5+22* x^4+1280*x^3-256*x^2-1024*x+512))/(512*x^16+4096*x^15+6144*x^14-28672*x^13 -79168*x^12+84736*x^11+359040*x^10-161536*x^9-917262*x^8+323072*x^7+143616 0*x^6-677888*x^5-1266688*x^4+917504*x^3+393216*x^2-524288*x+131072),x, alg orithm="fricas")
Output:
e^(-1/2*(16*x^9 + 64*x^8 - 32*x^7 - 320*x^6 + 11*x^5 + 646*x^4 - 128*x^3 - 512*x^2 + 256*x)/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256))
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).
Time = 0.60 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{\frac {- 16 x^{9} - 64 x^{8} + 32 x^{7} + 320 x^{6} - 11 x^{5} - 646 x^{4} + 128 x^{3} + 512 x^{2} - 256 x}{32 x^{8} + 128 x^{7} - 64 x^{6} - 640 x^{5} + 22 x^{4} + 1280 x^{3} - 256 x^{2} - 1024 x + 512}} \] Input:
integrate((-256*x**16-2048*x**15-3072*x**14+14336*x**13+39584*x**12-41984* x**11-178368*x**10+80384*x**9+456711*x**8-161536*x**7-721920*x**6+340480*x **5+642560*x**4-464896*x**3-196608*x**2+262144*x-65536)*exp((-16*x**9-64*x **8+32*x**7+320*x**6-11*x**5-646*x**4+128*x**3+512*x**2-256*x)/(32*x**8+12 8*x**7-64*x**6-640*x**5+22*x**4+1280*x**3-256*x**2-1024*x+512))/(512*x**16 +4096*x**15+6144*x**14-28672*x**13-79168*x**12+84736*x**11+359040*x**10-16 1536*x**9-917262*x**8+323072*x**7+1436160*x**6-677888*x**5-1266688*x**4+91 7504*x**3+393216*x**2-524288*x+131072),x)
Output:
exp((-16*x**9 - 64*x**8 + 32*x**7 + 320*x**6 - 11*x**5 - 646*x**4 + 128*x* *3 + 512*x**2 - 256*x)/(32*x**8 + 128*x**7 - 64*x**6 - 640*x**5 + 22*x**4 + 1280*x**3 - 256*x**2 - 1024*x + 512))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 15.60 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{\left (-\frac {3 \, x^{4}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} - \frac {1}{2} \, x\right )} \] Input:
integrate((-256*x^16-2048*x^15-3072*x^14+14336*x^13+39584*x^12-41984*x^11- 178368*x^10+80384*x^9+456711*x^8-161536*x^7-721920*x^6+340480*x^5+642560*x ^4-464896*x^3-196608*x^2+262144*x-65536)*exp((-16*x^9-64*x^8+32*x^7+320*x^ 6-11*x^5-646*x^4+128*x^3+512*x^2-256*x)/(32*x^8+128*x^7-64*x^6-640*x^5+22* x^4+1280*x^3-256*x^2-1024*x+512))/(512*x^16+4096*x^15+6144*x^14-28672*x^13 -79168*x^12+84736*x^11+359040*x^10-161536*x^9-917262*x^8+323072*x^7+143616 0*x^6-677888*x^5-1266688*x^4+917504*x^3+393216*x^2-524288*x+131072),x, alg orithm="maxima")
Output:
e^(-3*x^4/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) - 1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 423, normalized size of antiderivative = 12.82 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=e^{\left (-\frac {8 \, x^{9}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} - \frac {32 \, x^{8}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} + \frac {16 \, x^{7}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} + \frac {160 \, x^{6}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} - \frac {11 \, x^{5}}{2 \, {\left (16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256\right )}} - \frac {323 \, x^{4}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} + \frac {64 \, x^{3}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} + \frac {256 \, x^{2}}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256} - \frac {128 \, x}{16 \, x^{8} + 64 \, x^{7} - 32 \, x^{6} - 320 \, x^{5} + 11 \, x^{4} + 640 \, x^{3} - 128 \, x^{2} - 512 \, x + 256}\right )} \] Input:
integrate((-256*x^16-2048*x^15-3072*x^14+14336*x^13+39584*x^12-41984*x^11- 178368*x^10+80384*x^9+456711*x^8-161536*x^7-721920*x^6+340480*x^5+642560*x ^4-464896*x^3-196608*x^2+262144*x-65536)*exp((-16*x^9-64*x^8+32*x^7+320*x^ 6-11*x^5-646*x^4+128*x^3+512*x^2-256*x)/(32*x^8+128*x^7-64*x^6-640*x^5+22* x^4+1280*x^3-256*x^2-1024*x+512))/(512*x^16+4096*x^15+6144*x^14-28672*x^13 -79168*x^12+84736*x^11+359040*x^10-161536*x^9-917262*x^8+323072*x^7+143616 0*x^6-677888*x^5-1266688*x^4+917504*x^3+393216*x^2-524288*x+131072),x, alg orithm="giac")
Output:
e^(-8*x^9/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) - 32*x^8/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 64 0*x^3 - 128*x^2 - 512*x + 256) + 16*x^7/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^ 5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) + 160*x^6/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) - 11/2*x^5/( 16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) - 323*x^4/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 12 8*x^2 - 512*x + 256) + 64*x^3/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) + 256*x^2/(16*x^8 + 64*x^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256) - 128*x/(16*x^8 + 64*x ^7 - 32*x^6 - 320*x^5 + 11*x^4 + 640*x^3 - 128*x^2 - 512*x + 256))
Time = 0.54 (sec) , antiderivative size = 431, normalized size of antiderivative = 13.06 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx={\mathrm {e}}^{-\frac {128\,x}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{-\frac {8\,x^9}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{\frac {16\,x^7}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{-\frac {32\,x^8}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{\frac {64\,x^3}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{\frac {160\,x^6}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{\frac {256\,x^2}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{-\frac {323\,x^4}{16\,x^8+64\,x^7-32\,x^6-320\,x^5+11\,x^4+640\,x^3-128\,x^2-512\,x+256}}\,{\mathrm {e}}^{-\frac {11\,x^5}{32\,x^8+128\,x^7-64\,x^6-640\,x^5+22\,x^4+1280\,x^3-256\,x^2-1024\,x+512}} \] Input:
int(-(exp(-(256*x - 512*x^2 - 128*x^3 + 646*x^4 + 11*x^5 - 320*x^6 - 32*x^ 7 + 64*x^8 + 16*x^9)/(1280*x^3 - 256*x^2 - 1024*x + 22*x^4 - 640*x^5 - 64* x^6 + 128*x^7 + 32*x^8 + 512))*(196608*x^2 - 262144*x + 464896*x^3 - 64256 0*x^4 - 340480*x^5 + 721920*x^6 + 161536*x^7 - 456711*x^8 - 80384*x^9 + 17 8368*x^10 + 41984*x^11 - 39584*x^12 - 14336*x^13 + 3072*x^14 + 2048*x^15 + 256*x^16 + 65536))/(393216*x^2 - 524288*x + 917504*x^3 - 1266688*x^4 - 67 7888*x^5 + 1436160*x^6 + 323072*x^7 - 917262*x^8 - 161536*x^9 + 359040*x^1 0 + 84736*x^11 - 79168*x^12 - 28672*x^13 + 6144*x^14 + 4096*x^15 + 512*x^1 6 + 131072),x)
Output:
exp(-(128*x)/(640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x ^7 + 16*x^8 + 256))*exp(-(8*x^9)/(640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320 *x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp((16*x^7)/(640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp(-(32*x^8)/ (640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp((64*x^3)/(640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp((160*x^6)/(640*x^3 - 128*x^2 - 512*x + 11*x ^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp((256*x^2)/(640*x^3 - 1 28*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp(- (323*x^4)/(640*x^3 - 128*x^2 - 512*x + 11*x^4 - 320*x^5 - 32*x^6 + 64*x^7 + 16*x^8 + 256))*exp(-(11*x^5)/(1280*x^3 - 256*x^2 - 1024*x + 22*x^4 - 640 *x^5 - 64*x^6 + 128*x^7 + 32*x^8 + 512))
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.00 \[ \int \frac {e^{\frac {-256 x+512 x^2+128 x^3-646 x^4-11 x^5+320 x^6+32 x^7-64 x^8-16 x^9}{512-1024 x-256 x^2+1280 x^3+22 x^4-640 x^5-64 x^6+128 x^7+32 x^8}} \left (-65536+262144 x-196608 x^2-464896 x^3+642560 x^4+340480 x^5-721920 x^6-161536 x^7+456711 x^8+80384 x^9-178368 x^{10}-41984 x^{11}+39584 x^{12}+14336 x^{13}-3072 x^{14}-2048 x^{15}-256 x^{16}\right )}{131072-524288 x+393216 x^2+917504 x^3-1266688 x^4-677888 x^5+1436160 x^6+323072 x^7-917262 x^8-161536 x^9+359040 x^{10}+84736 x^{11}-79168 x^{12}-28672 x^{13}+6144 x^{14}+4096 x^{15}+512 x^{16}} \, dx=\frac {e^{\frac {144 x^{7}+96 x^{6}+1344 x^{3}+512}{16 x^{8}+64 x^{7}-32 x^{6}-320 x^{5}+11 x^{4}+640 x^{3}-128 x^{2}-512 x +256}}}{e^{\frac {16 x^{9}+1291 x^{5}+602 x^{4}+2304 x}{32 x^{8}+128 x^{7}-64 x^{6}-640 x^{5}+22 x^{4}+1280 x^{3}-256 x^{2}-1024 x +512}} e^{2}} \] Input:
int((-256*x^16-2048*x^15-3072*x^14+14336*x^13+39584*x^12-41984*x^11-178368 *x^10+80384*x^9+456711*x^8-161536*x^7-721920*x^6+340480*x^5+642560*x^4-464 896*x^3-196608*x^2+262144*x-65536)*exp((-16*x^9-64*x^8+32*x^7+320*x^6-11*x ^5-646*x^4+128*x^3+512*x^2-256*x)/(32*x^8+128*x^7-64*x^6-640*x^5+22*x^4+12 80*x^3-256*x^2-1024*x+512))/(512*x^16+4096*x^15+6144*x^14-28672*x^13-79168 *x^12+84736*x^11+359040*x^10-161536*x^9-917262*x^8+323072*x^7+1436160*x^6- 677888*x^5-1266688*x^4+917504*x^3+393216*x^2-524288*x+131072),x)
Output:
e**((144*x**7 + 96*x**6 + 1344*x**3 + 512)/(16*x**8 + 64*x**7 - 32*x**6 - 320*x**5 + 11*x**4 + 640*x**3 - 128*x**2 - 512*x + 256))/(e**((16*x**9 + 1 291*x**5 + 602*x**4 + 2304*x)/(32*x**8 + 128*x**7 - 64*x**6 - 640*x**5 + 2 2*x**4 + 1280*x**3 - 256*x**2 - 1024*x + 512))*e**2)