Integrand size = 404, antiderivative size = 25 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=e^{\frac {2 \left (3+\frac {1296}{(3+x)^8}\right )}{\left (3+e^{2 x}\right ) x}} \]
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).
Time = 0.51 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.36 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=e^{\frac {6 \left (6993+17496 x+20412 x^2+13608 x^3+5670 x^4+1512 x^5+252 x^6+24 x^7+x^8\right )}{\left (3+e^{2 x}\right ) x (3+x)^8}} \]
Integrate[(E^((41958 + 104976*x + 122472*x^2 + 81648*x^3 + 34020*x^4 + 907 2*x^5 + 1512*x^6 + 144*x^7 + 6*x^8)/(19683*x + 52488*x^2 + 61236*x^3 + 408 24*x^4 + 17010*x^5 + 4536*x^6 + 756*x^7 + 72*x^8 + 3*x^9 + E^(2*x)*(6561*x + 17496*x^2 + 20412*x^3 + 13608*x^4 + 5670*x^5 + 1512*x^6 + 252*x^7 + 24* x^8 + x^9)))*(-377622 - 1132866*x - 1417176*x^2 - 1102248*x^3 - 551124*x^4 - 183708*x^5 - 40824*x^6 - 5832*x^7 - 486*x^8 - 18*x^9 + E^(2*x)*(-125874 - 629370*x - 1186164*x^2 - 1312200*x^3 - 918540*x^4 - 428652*x^5 - 136080 *x^6 - 29160*x^7 - 4050*x^8 - 330*x^9 - 12*x^10)))/(177147*x^2 + 531441*x^ 3 + 708588*x^4 + 551124*x^5 + 275562*x^6 + 91854*x^7 + 20412*x^8 + 2916*x^ 9 + 243*x^10 + 9*x^11 + E^(4*x)*(19683*x^2 + 59049*x^3 + 78732*x^4 + 61236 *x^5 + 30618*x^6 + 10206*x^7 + 2268*x^8 + 324*x^9 + 27*x^10 + x^11) + E^(2 *x)*(118098*x^2 + 354294*x^3 + 472392*x^4 + 367416*x^5 + 183708*x^6 + 6123 6*x^7 + 13608*x^8 + 1944*x^9 + 162*x^10 + 6*x^11)),x]
E^((6*(6993 + 17496*x + 20412*x^2 + 13608*x^3 + 5670*x^4 + 1512*x^5 + 252* x^6 + 24*x^7 + x^8))/((3 + E^(2*x))*x*(3 + 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.
| \(\displaystyle \int \frac {\left (-18 x^9-486 x^8-5832 x^7-40824 x^6-183708 x^5-551124 x^4-1102248 x^3-1417176 x^2+e^{2 x} \left (-12 x^{10}-330 x^9-4050 x^8-29160 x^7-136080 x^6-428652 x^5-918540 x^4-1312200 x^3-1186164 x^2-629370 x-125874\right )-1132866 x-377622\right ) \exp \left (\frac {6 x^8+144 x^7+1512 x^6+9072 x^5+34020 x^4+81648 x^3+122472 x^2+104976 x+41958}{3 x^9+72 x^8+756 x^7+4536 x^6+17010 x^5+40824 x^4+61236 x^3+52488 x^2+e^{2 x} \left (x^9+24 x^8+252 x^7+1512 x^6+5670 x^5+13608 x^4+20412 x^3+17496 x^2+6561 x\right )+19683 x}\right )}{9 x^{11}+243 x^{10}+2916 x^9+20412 x^8+91854 x^7+275562 x^6+551124 x^5+708588 x^4+531441 x^3+177147 x^2+e^{4 x} \left (x^{11}+27 x^{10}+324 x^9+2268 x^8+10206 x^7+30618 x^6+61236 x^5+78732 x^4+59049 x^3+19683 x^2\right )+e^{2 x} \left (6 x^{11}+162 x^{10}+1944 x^9+13608 x^8+61236 x^7+183708 x^6+367416 x^5+472392 x^4+354294 x^3+118098 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 \left (-3 \left (x^9+27 x^8+324 x^7+2268 x^6+10206 x^5+30618 x^4+61236 x^3+78732 x^2+62937 x+20979\right )-e^{2 x} \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )\right ) \exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (e^{2 x}+3\right ) x (x+3)^8}\right )}{\left (e^{2 x}+3\right )^2 x^2 (x+3)^9}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int -\frac {\exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (3 \left (x^9+27 x^8+324 x^7+2268 x^6+10206 x^5+30618 x^4+61236 x^3+78732 x^2+62937 x+20979\right )+e^{2 x} \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )\right )}{\left (3+e^{2 x}\right )^2 x^2 (x+3)^9}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -6 \int \frac {\exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (3 \left (x^9+27 x^8+324 x^7+2268 x^6+10206 x^5+30618 x^4+61236 x^3+78732 x^2+62937 x+20979\right )+e^{2 x} \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )\right )}{\left (3+e^{2 x}\right )^2 x^2 (x+3)^9}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -6 \int \left (\frac {\exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )}{\left (3+e^{2 x}\right ) x^2 (x+3)^9}-\frac {6 \exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right )^2 x (x+3)^8}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -6 \int \left (\frac {6 \exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (-x^8-24 x^7-252 x^6-1512 x^5-5670 x^4-13608 x^3-20412 x^2-17496 x-6993\right )}{\left (3+e^{2 x}\right )^2 x (x+3)^8}+\frac {\exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )}{\left (3+e^{2 x}\right ) x^2 (x+3)^9}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -6 \int \left (\frac {6 \exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (-x^8-24 x^7-252 x^6-1512 x^5-5670 x^4-13608 x^3-20412 x^2-17496 x-6993\right )}{\left (3+e^{2 x}\right )^2 x (x+3)^8}+\frac {\exp \left (\frac {6 \left (x^8+24 x^7+252 x^6+1512 x^5+5670 x^4+13608 x^3+20412 x^2+17496 x+6993\right )}{\left (3+e^{2 x}\right ) x (x+3)^8}\right ) \left (2 x^{10}+55 x^9+675 x^8+4860 x^7+22680 x^6+71442 x^5+153090 x^4+218700 x^3+197694 x^2+104895 x+20979\right )}{\left (3+e^{2 x}\right ) x^2 (x+3)^9}\right )dx\) |
Int[(E^((41958 + 104976*x + 122472*x^2 + 81648*x^3 + 34020*x^4 + 9072*x^5 + 1512*x^6 + 144*x^7 + 6*x^8)/(19683*x + 52488*x^2 + 61236*x^3 + 40824*x^4 + 17010*x^5 + 4536*x^6 + 756*x^7 + 72*x^8 + 3*x^9 + E^(2*x)*(6561*x + 174 96*x^2 + 20412*x^3 + 13608*x^4 + 5670*x^5 + 1512*x^6 + 252*x^7 + 24*x^8 + x^9)))*(-377622 - 1132866*x - 1417176*x^2 - 1102248*x^3 - 551124*x^4 - 183 708*x^5 - 40824*x^6 - 5832*x^7 - 486*x^8 - 18*x^9 + E^(2*x)*(-125874 - 629 370*x - 1186164*x^2 - 1312200*x^3 - 918540*x^4 - 428652*x^5 - 136080*x^6 - 29160*x^7 - 4050*x^8 - 330*x^9 - 12*x^10)))/(177147*x^2 + 531441*x^3 + 70 8588*x^4 + 551124*x^5 + 275562*x^6 + 91854*x^7 + 20412*x^8 + 2916*x^9 + 24 3*x^10 + 9*x^11 + E^(4*x)*(19683*x^2 + 59049*x^3 + 78732*x^4 + 61236*x^5 + 30618*x^6 + 10206*x^7 + 2268*x^8 + 324*x^9 + 27*x^10 + x^11) + E^(2*x)*(1 18098*x^2 + 354294*x^3 + 472392*x^4 + 367416*x^5 + 183708*x^6 + 61236*x^7 + 13608*x^8 + 1944*x^9 + 162*x^10 + 6*x^11)),x]
3.6.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(23)=46\).
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32
\[{\mathrm e}^{\frac {6 x^{8}+144 x^{7}+1512 x^{6}+9072 x^{5}+34020 x^{4}+81648 x^{3}+122472 x^{2}+104976 x +41958}{x \left (3+x \right )^{8} \left ({\mathrm e}^{2 x}+3\right )}}\]
int(((-12*x^10-330*x^9-4050*x^8-29160*x^7-136080*x^6-428652*x^5-918540*x^4 -1312200*x^3-1186164*x^2-629370*x-125874)*exp(x)^2-18*x^9-486*x^8-5832*x^7 -40824*x^6-183708*x^5-551124*x^4-1102248*x^3-1417176*x^2-1132866*x-377622) *exp((6*x^8+144*x^7+1512*x^6+9072*x^5+34020*x^4+81648*x^3+122472*x^2+10497 6*x+41958)/((x^9+24*x^8+252*x^7+1512*x^6+5670*x^5+13608*x^4+20412*x^3+1749 6*x^2+6561*x)*exp(x)^2+3*x^9+72*x^8+756*x^7+4536*x^6+17010*x^5+40824*x^4+6 1236*x^3+52488*x^2+19683*x))/((x^11+27*x^10+324*x^9+2268*x^8+10206*x^7+306 18*x^6+61236*x^5+78732*x^4+59049*x^3+19683*x^2)*exp(x)^4+(6*x^11+162*x^10+ 1944*x^9+13608*x^8+61236*x^7+183708*x^6+367416*x^5+472392*x^4+354294*x^3+1 18098*x^2)*exp(x)^2+9*x^11+243*x^10+2916*x^9+20412*x^8+91854*x^7+275562*x^ 6+551124*x^5+708588*x^4+531441*x^3+177147*x^2),x)
exp(6*(x^8+24*x^7+252*x^6+1512*x^5+5670*x^4+13608*x^3+20412*x^2+17496*x+69 93)/x/(3+x)^8/(exp(2*x)+3))
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=e^{\left (\frac {6 \, {\left (x^{8} + 24 \, x^{7} + 252 \, x^{6} + 1512 \, x^{5} + 5670 \, x^{4} + 13608 \, x^{3} + 20412 \, x^{2} + 17496 \, x + 6993\right )}}{3 \, x^{9} + 72 \, x^{8} + 756 \, x^{7} + 4536 \, x^{6} + 17010 \, x^{5} + 40824 \, x^{4} + 61236 \, x^{3} + 52488 \, x^{2} + {\left (x^{9} + 24 \, x^{8} + 252 \, x^{7} + 1512 \, x^{6} + 5670 \, x^{5} + 13608 \, x^{4} + 20412 \, x^{3} + 17496 \, x^{2} + 6561 \, x\right )} e^{\left (2 \, x\right )} + 19683 \, x}\right )} \]
integrate(((-12*x^10-330*x^9-4050*x^8-29160*x^7-136080*x^6-428652*x^5-9185 40*x^4-1312200*x^3-1186164*x^2-629370*x-125874)*exp(x)^2-18*x^9-486*x^8-58 32*x^7-40824*x^6-183708*x^5-551124*x^4-1102248*x^3-1417176*x^2-1132866*x-3 77622)*exp((6*x^8+144*x^7+1512*x^6+9072*x^5+34020*x^4+81648*x^3+122472*x^2 +104976*x+41958)/((x^9+24*x^8+252*x^7+1512*x^6+5670*x^5+13608*x^4+20412*x^ 3+17496*x^2+6561*x)*exp(x)^2+3*x^9+72*x^8+756*x^7+4536*x^6+17010*x^5+40824 *x^4+61236*x^3+52488*x^2+19683*x))/((x^11+27*x^10+324*x^9+2268*x^8+10206*x ^7+30618*x^6+61236*x^5+78732*x^4+59049*x^3+19683*x^2)*exp(x)^4+(6*x^11+162 *x^10+1944*x^9+13608*x^8+61236*x^7+183708*x^6+367416*x^5+472392*x^4+354294 *x^3+118098*x^2)*exp(x)^2+9*x^11+243*x^10+2916*x^9+20412*x^8+91854*x^7+275 562*x^6+551124*x^5+708588*x^4+531441*x^3+177147*x^2),x, algorithm=\
e^(6*(x^8 + 24*x^7 + 252*x^6 + 1512*x^5 + 5670*x^4 + 13608*x^3 + 20412*x^2 + 17496*x + 6993)/(3*x^9 + 72*x^8 + 756*x^7 + 4536*x^6 + 17010*x^5 + 4082 4*x^4 + 61236*x^3 + 52488*x^2 + (x^9 + 24*x^8 + 252*x^7 + 1512*x^6 + 5670* x^5 + 13608*x^4 + 20412*x^3 + 17496*x^2 + 6561*x)*e^(2*x) + 19683*x))
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (17) = 34\).
Time = 1.77 (sec) , antiderivative size = 133, normalized size of antiderivative = 5.32 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=e^{\frac {6 x^{8} + 144 x^{7} + 1512 x^{6} + 9072 x^{5} + 34020 x^{4} + 81648 x^{3} + 122472 x^{2} + 104976 x + 41958}{3 x^{9} + 72 x^{8} + 756 x^{7} + 4536 x^{6} + 17010 x^{5} + 40824 x^{4} + 61236 x^{3} + 52488 x^{2} + 19683 x + \left (x^{9} + 24 x^{8} + 252 x^{7} + 1512 x^{6} + 5670 x^{5} + 13608 x^{4} + 20412 x^{3} + 17496 x^{2} + 6561 x\right ) e^{2 x}}} \]
integrate(((-12*x**10-330*x**9-4050*x**8-29160*x**7-136080*x**6-428652*x** 5-918540*x**4-1312200*x**3-1186164*x**2-629370*x-125874)*exp(x)**2-18*x**9 -486*x**8-5832*x**7-40824*x**6-183708*x**5-551124*x**4-1102248*x**3-141717 6*x**2-1132866*x-377622)*exp((6*x**8+144*x**7+1512*x**6+9072*x**5+34020*x* *4+81648*x**3+122472*x**2+104976*x+41958)/((x**9+24*x**8+252*x**7+1512*x** 6+5670*x**5+13608*x**4+20412*x**3+17496*x**2+6561*x)*exp(x)**2+3*x**9+72*x **8+756*x**7+4536*x**6+17010*x**5+40824*x**4+61236*x**3+52488*x**2+19683*x ))/((x**11+27*x**10+324*x**9+2268*x**8+10206*x**7+30618*x**6+61236*x**5+78 732*x**4+59049*x**3+19683*x**2)*exp(x)**4+(6*x**11+162*x**10+1944*x**9+136 08*x**8+61236*x**7+183708*x**6+367416*x**5+472392*x**4+354294*x**3+118098* x**2)*exp(x)**2+9*x**11+243*x**10+2916*x**9+20412*x**8+91854*x**7+275562*x **6+551124*x**5+708588*x**4+531441*x**3+177147*x**2),x)
exp((6*x**8 + 144*x**7 + 1512*x**6 + 9072*x**5 + 34020*x**4 + 81648*x**3 + 122472*x**2 + 104976*x + 41958)/(3*x**9 + 72*x**8 + 756*x**7 + 4536*x**6 + 17010*x**5 + 40824*x**4 + 61236*x**3 + 52488*x**2 + 19683*x + (x**9 + 24 *x**8 + 252*x**7 + 1512*x**6 + 5670*x**5 + 13608*x**4 + 20412*x**3 + 17496 *x**2 + 6561*x)*exp(2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (23) = 46\).
Time = 195.69 (sec) , antiderivative size = 432, normalized size of antiderivative = 17.28 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=e^{\left (-\frac {864}{3 \, x^{8} + 72 \, x^{7} + 756 \, x^{6} + 4536 \, x^{5} + 17010 \, x^{4} + 40824 \, x^{3} + 61236 \, x^{2} + {\left (x^{8} + 24 \, x^{7} + 252 \, x^{6} + 1512 \, x^{5} + 5670 \, x^{4} + 13608 \, x^{3} + 20412 \, x^{2} + 17496 \, x + 6561\right )} e^{\left (2 \, x\right )} + 52488 \, x + 19683} - \frac {288}{3 \, x^{7} + 63 \, x^{6} + 567 \, x^{5} + 2835 \, x^{4} + 8505 \, x^{3} + 15309 \, x^{2} + {\left (x^{7} + 21 \, x^{6} + 189 \, x^{5} + 945 \, x^{4} + 2835 \, x^{3} + 5103 \, x^{2} + 5103 \, x + 2187\right )} e^{\left (2 \, x\right )} + 15309 \, x + 6561} - \frac {96}{3 \, x^{6} + 54 \, x^{5} + 405 \, x^{4} + 1620 \, x^{3} + 3645 \, x^{2} + {\left (x^{6} + 18 \, x^{5} + 135 \, x^{4} + 540 \, x^{3} + 1215 \, x^{2} + 1458 \, x + 729\right )} e^{\left (2 \, x\right )} + 4374 \, x + 2187} - \frac {32}{3 \, x^{5} + 45 \, x^{4} + 270 \, x^{3} + 810 \, x^{2} + {\left (x^{5} + 15 \, x^{4} + 90 \, x^{3} + 270 \, x^{2} + 405 \, x + 243\right )} e^{\left (2 \, x\right )} + 1215 \, x + 729} - \frac {32}{3 \, {\left (3 \, x^{4} + 36 \, x^{3} + 162 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} e^{\left (2 \, x\right )} + 324 \, x + 243\right )}} - \frac {32}{9 \, {\left (3 \, x^{3} + 27 \, x^{2} + {\left (x^{3} + 9 \, x^{2} + 27 \, x + 27\right )} e^{\left (2 \, x\right )} + 81 \, x + 81\right )}} - \frac {32}{27 \, {\left (3 \, x^{2} + {\left (x^{2} + 6 \, x + 9\right )} e^{\left (2 \, x\right )} + 18 \, x + 27\right )}} - \frac {32}{81 \, {\left ({\left (x + 3\right )} e^{\left (2 \, x\right )} + 3 \, x + 9\right )}} + \frac {518}{81 \, {\left (x e^{\left (2 \, x\right )} + 3 \, x\right )}}\right )} \]
integrate(((-12*x^10-330*x^9-4050*x^8-29160*x^7-136080*x^6-428652*x^5-9185 40*x^4-1312200*x^3-1186164*x^2-629370*x-125874)*exp(x)^2-18*x^9-486*x^8-58 32*x^7-40824*x^6-183708*x^5-551124*x^4-1102248*x^3-1417176*x^2-1132866*x-3 77622)*exp((6*x^8+144*x^7+1512*x^6+9072*x^5+34020*x^4+81648*x^3+122472*x^2 +104976*x+41958)/((x^9+24*x^8+252*x^7+1512*x^6+5670*x^5+13608*x^4+20412*x^ 3+17496*x^2+6561*x)*exp(x)^2+3*x^9+72*x^8+756*x^7+4536*x^6+17010*x^5+40824 *x^4+61236*x^3+52488*x^2+19683*x))/((x^11+27*x^10+324*x^9+2268*x^8+10206*x ^7+30618*x^6+61236*x^5+78732*x^4+59049*x^3+19683*x^2)*exp(x)^4+(6*x^11+162 *x^10+1944*x^9+13608*x^8+61236*x^7+183708*x^6+367416*x^5+472392*x^4+354294 *x^3+118098*x^2)*exp(x)^2+9*x^11+243*x^10+2916*x^9+20412*x^8+91854*x^7+275 562*x^6+551124*x^5+708588*x^4+531441*x^3+177147*x^2),x, algorithm=\
e^(-864/(3*x^8 + 72*x^7 + 756*x^6 + 4536*x^5 + 17010*x^4 + 40824*x^3 + 612 36*x^2 + (x^8 + 24*x^7 + 252*x^6 + 1512*x^5 + 5670*x^4 + 13608*x^3 + 20412 *x^2 + 17496*x + 6561)*e^(2*x) + 52488*x + 19683) - 288/(3*x^7 + 63*x^6 + 567*x^5 + 2835*x^4 + 8505*x^3 + 15309*x^2 + (x^7 + 21*x^6 + 189*x^5 + 945* x^4 + 2835*x^3 + 5103*x^2 + 5103*x + 2187)*e^(2*x) + 15309*x + 6561) - 96/ (3*x^6 + 54*x^5 + 405*x^4 + 1620*x^3 + 3645*x^2 + (x^6 + 18*x^5 + 135*x^4 + 540*x^3 + 1215*x^2 + 1458*x + 729)*e^(2*x) + 4374*x + 2187) - 32/(3*x^5 + 45*x^4 + 270*x^3 + 810*x^2 + (x^5 + 15*x^4 + 90*x^3 + 270*x^2 + 405*x + 243)*e^(2*x) + 1215*x + 729) - 32/3/(3*x^4 + 36*x^3 + 162*x^2 + (x^4 + 12* x^3 + 54*x^2 + 108*x + 81)*e^(2*x) + 324*x + 243) - 32/9/(3*x^3 + 27*x^2 + (x^3 + 9*x^2 + 27*x + 27)*e^(2*x) + 81*x + 81) - 32/27/(3*x^2 + (x^2 + 6* x + 9)*e^(2*x) + 18*x + 27) - 32/81/((x + 3)*e^(2*x) + 3*x + 9) + 518/81/( x*e^(2*x) + 3*x))
Exception generated. \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((-12*x^10-330*x^9-4050*x^8-29160*x^7-136080*x^6-428652*x^5-9185 40*x^4-1312200*x^3-1186164*x^2-629370*x-125874)*exp(x)^2-18*x^9-486*x^8-58 32*x^7-40824*x^6-183708*x^5-551124*x^4-1102248*x^3-1417176*x^2-1132866*x-3 77622)*exp((6*x^8+144*x^7+1512*x^6+9072*x^5+34020*x^4+81648*x^3+122472*x^2 +104976*x+41958)/((x^9+24*x^8+252*x^7+1512*x^6+5670*x^5+13608*x^4+20412*x^ 3+17496*x^2+6561*x)*exp(x)^2+3*x^9+72*x^8+756*x^7+4536*x^6+17010*x^5+40824 *x^4+61236*x^3+52488*x^2+19683*x))/((x^11+27*x^10+324*x^9+2268*x^8+10206*x ^7+30618*x^6+61236*x^5+78732*x^4+59049*x^3+19683*x^2)*exp(x)^4+(6*x^11+162 *x^10+1944*x^9+13608*x^8+61236*x^7+183708*x^6+367416*x^5+472392*x^4+354294 *x^3+118098*x^2)*exp(x)^2+9*x^11+243*x^10+2916*x^9+20412*x^8+91854*x^7+275 562*x^6+551124*x^5+708588*x^4+531441*x^3+177147*x^2),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-864,[1,61]%%%}+%%%{-140832,[1,60]%%%}+%%%{-11267640,[1,59 ]%%%}+%%%
Time = 9.15 (sec) , antiderivative size = 1107, normalized size of antiderivative = 44.28 \[ \int \frac {e^{\frac {41958+104976 x+122472 x^2+81648 x^3+34020 x^4+9072 x^5+1512 x^6+144 x^7+6 x^8}{19683 x+52488 x^2+61236 x^3+40824 x^4+17010 x^5+4536 x^6+756 x^7+72 x^8+3 x^9+e^{2 x} \left (6561 x+17496 x^2+20412 x^3+13608 x^4+5670 x^5+1512 x^6+252 x^7+24 x^8+x^9\right )}} \left (-377622-1132866 x-1417176 x^2-1102248 x^3-551124 x^4-183708 x^5-40824 x^6-5832 x^7-486 x^8-18 x^9+e^{2 x} \left (-125874-629370 x-1186164 x^2-1312200 x^3-918540 x^4-428652 x^5-136080 x^6-29160 x^7-4050 x^8-330 x^9-12 x^{10}\right )\right )}{177147 x^2+531441 x^3+708588 x^4+551124 x^5+275562 x^6+91854 x^7+20412 x^8+2916 x^9+243 x^{10}+9 x^{11}+e^{4 x} \left (19683 x^2+59049 x^3+78732 x^4+61236 x^5+30618 x^6+10206 x^7+2268 x^8+324 x^9+27 x^{10}+x^{11}\right )+e^{2 x} \left (118098 x^2+354294 x^3+472392 x^4+367416 x^5+183708 x^6+61236 x^7+13608 x^8+1944 x^9+162 x^{10}+6 x^{11}\right )} \, dx=\text {Too large to display} \]
int(-(exp((104976*x + 122472*x^2 + 81648*x^3 + 34020*x^4 + 9072*x^5 + 1512 *x^6 + 144*x^7 + 6*x^8 + 41958)/(19683*x + exp(2*x)*(6561*x + 17496*x^2 + 20412*x^3 + 13608*x^4 + 5670*x^5 + 1512*x^6 + 252*x^7 + 24*x^8 + x^9) + 52 488*x^2 + 61236*x^3 + 40824*x^4 + 17010*x^5 + 4536*x^6 + 756*x^7 + 72*x^8 + 3*x^9))*(1132866*x + exp(2*x)*(629370*x + 1186164*x^2 + 1312200*x^3 + 91 8540*x^4 + 428652*x^5 + 136080*x^6 + 29160*x^7 + 4050*x^8 + 330*x^9 + 12*x ^10 + 125874) + 1417176*x^2 + 1102248*x^3 + 551124*x^4 + 183708*x^5 + 4082 4*x^6 + 5832*x^7 + 486*x^8 + 18*x^9 + 377622))/(exp(2*x)*(118098*x^2 + 354 294*x^3 + 472392*x^4 + 367416*x^5 + 183708*x^6 + 61236*x^7 + 13608*x^8 + 1 944*x^9 + 162*x^10 + 6*x^11) + exp(4*x)*(19683*x^2 + 59049*x^3 + 78732*x^4 + 61236*x^5 + 30618*x^6 + 10206*x^7 + 2268*x^8 + 324*x^9 + 27*x^10 + x^11 ) + 177147*x^2 + 531441*x^3 + 708588*x^4 + 551124*x^5 + 275562*x^6 + 91854 *x^7 + 20412*x^8 + 2916*x^9 + 243*x^10 + 9*x^11),x)
exp((122472*x)/(52488*x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp (2*x) + 13608*x^3*exp(2*x) + 5670*x^4*exp(2*x) + 1512*x^5*exp(2*x) + 252*x ^6*exp(2*x) + 24*x^7*exp(2*x) + x^8*exp(2*x) + 61236*x^2 + 40824*x^3 + 170 10*x^4 + 4536*x^5 + 756*x^6 + 72*x^7 + 3*x^8 + 19683))*exp((6*x^7)/(52488* x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp(2*x) + 13608*x^3*exp( 2*x) + 5670*x^4*exp(2*x) + 1512*x^5*exp(2*x) + 252*x^6*exp(2*x) + 24*x^7*e xp(2*x) + x^8*exp(2*x) + 61236*x^2 + 40824*x^3 + 17010*x^4 + 4536*x^5 + 75 6*x^6 + 72*x^7 + 3*x^8 + 19683))*exp((144*x^6)/(52488*x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp(2*x) + 13608*x^3*exp(2*x) + 5670*x^4*exp( 2*x) + 1512*x^5*exp(2*x) + 252*x^6*exp(2*x) + 24*x^7*exp(2*x) + x^8*exp(2* x) + 61236*x^2 + 40824*x^3 + 17010*x^4 + 4536*x^5 + 756*x^6 + 72*x^7 + 3*x ^8 + 19683))*exp((1512*x^5)/(52488*x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp(2*x) + 13608*x^3*exp(2*x) + 5670*x^4*exp(2*x) + 1512*x^5*exp (2*x) + 252*x^6*exp(2*x) + 24*x^7*exp(2*x) + x^8*exp(2*x) + 61236*x^2 + 40 824*x^3 + 17010*x^4 + 4536*x^5 + 756*x^6 + 72*x^7 + 3*x^8 + 19683))*exp((9 072*x^4)/(52488*x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp(2*x) + 13608*x^3*exp(2*x) + 5670*x^4*exp(2*x) + 1512*x^5*exp(2*x) + 252*x^6*exp (2*x) + 24*x^7*exp(2*x) + x^8*exp(2*x) + 61236*x^2 + 40824*x^3 + 17010*x^4 + 4536*x^5 + 756*x^6 + 72*x^7 + 3*x^8 + 19683))*exp((34020*x^3)/(52488*x + 6561*exp(2*x) + 17496*x*exp(2*x) + 20412*x^2*exp(2*x) + 13608*x^3*exp...