Integrand size = 398, antiderivative size = 30 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=e^{-4+4 \left (-x+\frac {9}{\left (-3+\left (3-e^x\right ) x (-5+2 x)\right )^2}\right )} \end {dmath*}
Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=e^{-4-4 x+\frac {36}{\left (3-5 \left (-3+e^x\right ) x+2 \left (-3+e^x\right ) x^2\right )^2}} \end {dmath*}
Integrate[(E^((2*(-198*x - 558*x^2 - 18*x^3 + 288*x^4 - 72*x^5 + E^(2*x)*( -50*x^2 - 10*x^3 + 32*x^4 - 8*x^5) + E^x*(60*x + 336*x^2 + 36*x^3 - 192*x^ 4 + 48*x^5)))/(9 + 90*x + 189*x^2 - 180*x^3 + 36*x^4 + E^x*(-30*x - 138*x^ 2 + 120*x^3 - 24*x^4) + E^(2*x)*(25*x^2 - 20*x^3 + 4*x^4)))*(-1188 - 756*x - 7452*x^2 - 7020*x^3 + 14904*x^4 - 6480*x^5 + 864*x^6 + E^x*(360 + 612*x + 5040*x^2 + 9180*x^3 - 15336*x^4 + 6480*x^5 - 864*x^6) + E^(3*x)*(500*x^ 3 - 600*x^4 + 240*x^5 - 32*x^6) + E^(2*x)*(-900*x^2 - 3780*x^3 + 5256*x^4 - 2160*x^5 + 288*x^6)))/(27 + 405*x + 1863*x^2 + 1755*x^3 - 3726*x^4 + 162 0*x^5 - 216*x^6 + E^(2*x)*(225*x^2 + 945*x^3 - 1314*x^4 + 540*x^5 - 72*x^6 ) + E^(3*x)*(-125*x^3 + 150*x^4 - 60*x^5 + 8*x^6) + E^x*(-135*x - 1296*x^2 - 2295*x^3 + 3834*x^4 - 1620*x^5 + 216*x^6)),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.
| \(\displaystyle \int \frac {\left (864 x^6-6480 x^5+14904 x^4-7020 x^3-7452 x^2+e^{3 x} \left (-32 x^6+240 x^5-600 x^4+500 x^3\right )+e^x \left (-864 x^6+6480 x^5-15336 x^4+9180 x^3+5040 x^2+612 x+360\right )+e^{2 x} \left (288 x^6-2160 x^5+5256 x^4-3780 x^3-900 x^2\right )-756 x-1188\right ) \exp \left (\frac {2 \left (-72 x^5+288 x^4-18 x^3-558 x^2+e^{2 x} \left (-8 x^5+32 x^4-10 x^3-50 x^2\right )+e^x \left (48 x^5-192 x^4+36 x^3+336 x^2+60 x\right )-198 x\right )}{36 x^4-180 x^3+189 x^2+e^x \left (-24 x^4+120 x^3-138 x^2-30 x\right )+e^{2 x} \left (4 x^4-20 x^3+25 x^2\right )+90 x+9}\right )}{-216 x^6+1620 x^5-3726 x^4+1755 x^3+1863 x^2+e^{3 x} \left (8 x^6-60 x^5+150 x^4-125 x^3\right )+e^{2 x} \left (-72 x^6+540 x^5-1314 x^4+945 x^3+225 x^2\right )+e^x \left (216 x^6-1620 x^5+3834 x^4-2295 x^3-1296 x^2-135 x\right )+405 x+27} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right ) \exp \left (-\frac {4 x \left (9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )+e^{2 x} x (x+1) (5-2 x)^2\right )}{\left (2 \left (e^x-3\right ) x^2-5 \left (e^x-3\right ) x+3\right )^2}\right )}{\left (2 \left (e^x-3\right ) x^2-5 \left (e^x-3\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (-2 \left (3-e^x\right ) x^2+5 \left (3-e^x\right ) x+3\right )^2}\right ) \left (e^{3 x} (5-2 x)^3 x^3-9 e^{2 x} (5-2 x)^2 \left (-2 x^2+5 x+1\right ) x^2+9 e^x \left (-24 x^6+180 x^5-426 x^4+255 x^3+140 x^2+17 x+10\right )-27 \left (-8 x^6+60 x^5-138 x^4+65 x^3+69 x^2+7 x+11\right )\right )}{\left (-2 \left (3-e^x\right ) x^2+5 \left (3-e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (-2 \left (3-e^x\right ) x^2+5 \left (3-e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (-2 \left (3-e^x\right ) x^2+5 \left (3-e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (-2 \left (3-e^x\right ) x^2+5 \left (3-e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {18 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (2 x^2-x-5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^2}-\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right )-\frac {54 \exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (4 x^4-20 x^3+23 x^2+x+5\right )}{x (2 x-5) \left (2 e^x x^2-6 x^2-5 e^x x+15 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {\exp \left (-\frac {4 x \left (e^{2 x} x (x+1) (5-2 x)^2+9 \left (4 x^4-16 x^3+x^2+31 x+11\right )-6 e^x \left (4 x^4-16 x^3+3 x^2+28 x+5\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^2}\right ) \left (-e^{3 x} x^3 (2 x-5)^3+9 e^{2 x} (5-2 x)^2 x^2 \left (2 x^2-5 x-1\right )+27 \left (8 x^6-60 x^5+138 x^4-65 x^3-69 x^2-7 x-11\right )-9 e^x \left (24 x^6-180 x^5+426 x^4-255 x^3-140 x^2-17 x-10\right )\right )}{\left (2 \left (-3+e^x\right ) x^2-5 \left (-3+e^x\right ) x+3\right )^3}dx\) |
Int[(E^((2*(-198*x - 558*x^2 - 18*x^3 + 288*x^4 - 72*x^5 + E^(2*x)*(-50*x^ 2 - 10*x^3 + 32*x^4 - 8*x^5) + E^x*(60*x + 336*x^2 + 36*x^3 - 192*x^4 + 48 *x^5)))/(9 + 90*x + 189*x^2 - 180*x^3 + 36*x^4 + E^x*(-30*x - 138*x^2 + 12 0*x^3 - 24*x^4) + E^(2*x)*(25*x^2 - 20*x^3 + 4*x^4)))*(-1188 - 756*x - 745 2*x^2 - 7020*x^3 + 14904*x^4 - 6480*x^5 + 864*x^6 + E^x*(360 + 612*x + 504 0*x^2 + 9180*x^3 - 15336*x^4 + 6480*x^5 - 864*x^6) + E^(3*x)*(500*x^3 - 60 0*x^4 + 240*x^5 - 32*x^6) + E^(2*x)*(-900*x^2 - 3780*x^3 + 5256*x^4 - 2160 *x^5 + 288*x^6)))/(27 + 405*x + 1863*x^2 + 1755*x^3 - 3726*x^4 + 1620*x^5 - 216*x^6 + E^(2*x)*(225*x^2 + 945*x^3 - 1314*x^4 + 540*x^5 - 72*x^6) + E^ (3*x)*(-125*x^3 + 150*x^4 - 60*x^5 + 8*x^6) + E^x*(-135*x - 1296*x^2 - 229 5*x^3 + 3834*x^4 - 1620*x^5 + 216*x^6)),x]
3.4.27.3.1 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. \(156\) vs. \(2(27)=54\).
Time = 65.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 5.23
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {2 \left (-8 x^{5}+32 x^{4}-10 x^{3}-50 x^{2}\right ) {\mathrm e}^{2 x}+2 \left (48 x^{5}-192 x^{4}+36 x^{3}+336 x^{2}+60 x \right ) {\mathrm e}^{x}-144 x^{5}+576 x^{4}-36 x^{3}-1116 x^{2}-396 x}{4 \,{\mathrm e}^{2 x} x^{4}-20 \,{\mathrm e}^{2 x} x^{3}-24 \,{\mathrm e}^{x} x^{4}+25 \,{\mathrm e}^{2 x} x^{2}+120 \,{\mathrm e}^{x} x^{3}+36 x^{4}-138 \,{\mathrm e}^{x} x^{2}-180 x^{3}-30 \,{\mathrm e}^{x} x +189 x^{2}+90 x +9}}\) | \(157\) |
int(((-32*x^6+240*x^5-600*x^4+500*x^3)*exp(x)^3+(288*x^6-2160*x^5+5256*x^4 -3780*x^3-900*x^2)*exp(x)^2+(-864*x^6+6480*x^5-15336*x^4+9180*x^3+5040*x^2 +612*x+360)*exp(x)+864*x^6-6480*x^5+14904*x^4-7020*x^3-7452*x^2-756*x-1188 )*exp(((-8*x^5+32*x^4-10*x^3-50*x^2)*exp(x)^2+(48*x^5-192*x^4+36*x^3+336*x ^2+60*x)*exp(x)-72*x^5+288*x^4-18*x^3-558*x^2-198*x)/((4*x^4-20*x^3+25*x^2 )*exp(x)^2+(-24*x^4+120*x^3-138*x^2-30*x)*exp(x)+36*x^4-180*x^3+189*x^2+90 *x+9))^2/((8*x^6-60*x^5+150*x^4-125*x^3)*exp(x)^3+(-72*x^6+540*x^5-1314*x^ 4+945*x^3+225*x^2)*exp(x)^2+(216*x^6-1620*x^5+3834*x^4-2295*x^3-1296*x^2-1 35*x)*exp(x)-216*x^6+1620*x^5-3726*x^4+1755*x^3+1863*x^2+405*x+27),x,metho d=_RETURNVERBOSE)
exp(((-8*x^5+32*x^4-10*x^3-50*x^2)*exp(x)^2+(48*x^5-192*x^4+36*x^3+336*x^2 +60*x)*exp(x)-72*x^5+288*x^4-18*x^3-558*x^2-198*x)/(4*exp(x)^2*x^4-20*exp( x)^2*x^3-24*exp(x)*x^4+25*exp(x)^2*x^2+120*exp(x)*x^3+36*x^4-138*exp(x)*x^ 2-180*x^3-30*exp(x)*x+189*x^2+90*x+9))^2
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.90 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=e^{\left (-\frac {4 \, {\left (36 \, x^{5} - 144 \, x^{4} + 9 \, x^{3} + 279 \, x^{2} + {\left (4 \, x^{5} - 16 \, x^{4} + 5 \, x^{3} + 25 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (4 \, x^{5} - 16 \, x^{4} + 3 \, x^{3} + 28 \, x^{2} + 5 \, x\right )} e^{x} + 99 \, x\right )}}{36 \, x^{4} - 180 \, x^{3} + 189 \, x^{2} + {\left (4 \, x^{4} - 20 \, x^{3} + 25 \, x^{2}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (4 \, x^{4} - 20 \, x^{3} + 23 \, x^{2} + 5 \, x\right )} e^{x} + 90 \, x + 9}\right )} \end {dmath*}
integrate(((-32*x^6+240*x^5-600*x^4+500*x^3)*exp(x)^3+(288*x^6-2160*x^5+52 56*x^4-3780*x^3-900*x^2)*exp(x)^2+(-864*x^6+6480*x^5-15336*x^4+9180*x^3+50 40*x^2+612*x+360)*exp(x)+864*x^6-6480*x^5+14904*x^4-7020*x^3-7452*x^2-756* x-1188)*exp(((-8*x^5+32*x^4-10*x^3-50*x^2)*exp(x)^2+(48*x^5-192*x^4+36*x^3 +336*x^2+60*x)*exp(x)-72*x^5+288*x^4-18*x^3-558*x^2-198*x)/((4*x^4-20*x^3+ 25*x^2)*exp(x)^2+(-24*x^4+120*x^3-138*x^2-30*x)*exp(x)+36*x^4-180*x^3+189* x^2+90*x+9))^2/((8*x^6-60*x^5+150*x^4-125*x^3)*exp(x)^3+(-72*x^6+540*x^5-1 314*x^4+945*x^3+225*x^2)*exp(x)^2+(216*x^6-1620*x^5+3834*x^4-2295*x^3-1296 *x^2-135*x)*exp(x)-216*x^6+1620*x^5-3726*x^4+1755*x^3+1863*x^2+405*x+27),x , algorithm=\
e^(-4*(36*x^5 - 144*x^4 + 9*x^3 + 279*x^2 + (4*x^5 - 16*x^4 + 5*x^3 + 25*x ^2)*e^(2*x) - 6*(4*x^5 - 16*x^4 + 3*x^3 + 28*x^2 + 5*x)*e^x + 99*x)/(36*x^ 4 - 180*x^3 + 189*x^2 + (4*x^4 - 20*x^3 + 25*x^2)*e^(2*x) - 6*(4*x^4 - 20* x^3 + 23*x^2 + 5*x)*e^x + 90*x + 9))
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (22) = 44\).
Time = 3.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.77 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=e^{\frac {2 \left (- 72 x^{5} + 288 x^{4} - 18 x^{3} - 558 x^{2} - 198 x + \left (- 8 x^{5} + 32 x^{4} - 10 x^{3} - 50 x^{2}\right ) e^{2 x} + \left (48 x^{5} - 192 x^{4} + 36 x^{3} + 336 x^{2} + 60 x\right ) e^{x}\right )}{36 x^{4} - 180 x^{3} + 189 x^{2} + 90 x + \left (4 x^{4} - 20 x^{3} + 25 x^{2}\right ) e^{2 x} + \left (- 24 x^{4} + 120 x^{3} - 138 x^{2} - 30 x\right ) e^{x} + 9}} \end {dmath*}
integrate(((-32*x**6+240*x**5-600*x**4+500*x**3)*exp(x)**3+(288*x**6-2160* x**5+5256*x**4-3780*x**3-900*x**2)*exp(x)**2+(-864*x**6+6480*x**5-15336*x* *4+9180*x**3+5040*x**2+612*x+360)*exp(x)+864*x**6-6480*x**5+14904*x**4-702 0*x**3-7452*x**2-756*x-1188)*exp(((-8*x**5+32*x**4-10*x**3-50*x**2)*exp(x) **2+(48*x**5-192*x**4+36*x**3+336*x**2+60*x)*exp(x)-72*x**5+288*x**4-18*x* *3-558*x**2-198*x)/((4*x**4-20*x**3+25*x**2)*exp(x)**2+(-24*x**4+120*x**3- 138*x**2-30*x)*exp(x)+36*x**4-180*x**3+189*x**2+90*x+9))**2/((8*x**6-60*x* *5+150*x**4-125*x**3)*exp(x)**3+(-72*x**6+540*x**5-1314*x**4+945*x**3+225* x**2)*exp(x)**2+(216*x**6-1620*x**5+3834*x**4-2295*x**3-1296*x**2-135*x)*e xp(x)-216*x**6+1620*x**5-3726*x**4+1755*x**3+1863*x**2+405*x+27),x)
exp(2*(-72*x**5 + 288*x**4 - 18*x**3 - 558*x**2 - 198*x + (-8*x**5 + 32*x* *4 - 10*x**3 - 50*x**2)*exp(2*x) + (48*x**5 - 192*x**4 + 36*x**3 + 336*x** 2 + 60*x)*exp(x))/(36*x**4 - 180*x**3 + 189*x**2 + 90*x + (4*x**4 - 20*x** 3 + 25*x**2)*exp(2*x) + (-24*x**4 + 120*x**3 - 138*x**2 - 30*x)*exp(x) + 9 ))
Timed out. \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=\text {Timed out} \end {dmath*}
integrate(((-32*x^6+240*x^5-600*x^4+500*x^3)*exp(x)^3+(288*x^6-2160*x^5+52 56*x^4-3780*x^3-900*x^2)*exp(x)^2+(-864*x^6+6480*x^5-15336*x^4+9180*x^3+50 40*x^2+612*x+360)*exp(x)+864*x^6-6480*x^5+14904*x^4-7020*x^3-7452*x^2-756* x-1188)*exp(((-8*x^5+32*x^4-10*x^3-50*x^2)*exp(x)^2+(48*x^5-192*x^4+36*x^3 +336*x^2+60*x)*exp(x)-72*x^5+288*x^4-18*x^3-558*x^2-198*x)/((4*x^4-20*x^3+ 25*x^2)*exp(x)^2+(-24*x^4+120*x^3-138*x^2-30*x)*exp(x)+36*x^4-180*x^3+189* x^2+90*x+9))^2/((8*x^6-60*x^5+150*x^4-125*x^3)*exp(x)^3+(-72*x^6+540*x^5-1 314*x^4+945*x^3+225*x^2)*exp(x)^2+(216*x^6-1620*x^5+3834*x^4-2295*x^3-1296 *x^2-135*x)*exp(x)-216*x^6+1620*x^5-3726*x^4+1755*x^3+1863*x^2+405*x+27),x , algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (23) = 46\).
Time = 24.39 (sec) , antiderivative size = 175, normalized size of antiderivative = 5.83 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=e^{\left (-3 \, x - \frac {4 \, x^{5} e^{\left (2 \, x\right )} - 24 \, x^{5} e^{x} + 36 \, x^{5} - 4 \, x^{4} e^{\left (2 \, x\right )} + 24 \, x^{4} e^{x} - 36 \, x^{4} - 55 \, x^{3} e^{\left (2 \, x\right )} + 342 \, x^{3} e^{x} - 531 \, x^{3} + 100 \, x^{2} e^{\left (2 \, x\right )} - 582 \, x^{2} e^{x} + 846 \, x^{2} - 120 \, x e^{x} + 369 \, x}{4 \, x^{4} e^{\left (2 \, x\right )} - 24 \, x^{4} e^{x} + 36 \, x^{4} - 20 \, x^{3} e^{\left (2 \, x\right )} + 120 \, x^{3} e^{x} - 180 \, x^{3} + 25 \, x^{2} e^{\left (2 \, x\right )} - 138 \, x^{2} e^{x} + 189 \, x^{2} - 30 \, x e^{x} + 90 \, x + 9}\right )} \end {dmath*}
integrate(((-32*x^6+240*x^5-600*x^4+500*x^3)*exp(x)^3+(288*x^6-2160*x^5+52 56*x^4-3780*x^3-900*x^2)*exp(x)^2+(-864*x^6+6480*x^5-15336*x^4+9180*x^3+50 40*x^2+612*x+360)*exp(x)+864*x^6-6480*x^5+14904*x^4-7020*x^3-7452*x^2-756* x-1188)*exp(((-8*x^5+32*x^4-10*x^3-50*x^2)*exp(x)^2+(48*x^5-192*x^4+36*x^3 +336*x^2+60*x)*exp(x)-72*x^5+288*x^4-18*x^3-558*x^2-198*x)/((4*x^4-20*x^3+ 25*x^2)*exp(x)^2+(-24*x^4+120*x^3-138*x^2-30*x)*exp(x)+36*x^4-180*x^3+189* x^2+90*x+9))^2/((8*x^6-60*x^5+150*x^4-125*x^3)*exp(x)^3+(-72*x^6+540*x^5-1 314*x^4+945*x^3+225*x^2)*exp(x)^2+(216*x^6-1620*x^5+3834*x^4-2295*x^3-1296 *x^2-135*x)*exp(x)-216*x^6+1620*x^5-3726*x^4+1755*x^3+1863*x^2+405*x+27),x , algorithm=\
e^(-3*x - (4*x^5*e^(2*x) - 24*x^5*e^x + 36*x^5 - 4*x^4*e^(2*x) + 24*x^4*e^ x - 36*x^4 - 55*x^3*e^(2*x) + 342*x^3*e^x - 531*x^3 + 100*x^2*e^(2*x) - 58 2*x^2*e^x + 846*x^2 - 120*x*e^x + 369*x)/(4*x^4*e^(2*x) - 24*x^4*e^x + 36* x^4 - 20*x^3*e^(2*x) + 120*x^3*e^x - 180*x^3 + 25*x^2*e^(2*x) - 138*x^2*e^ x + 189*x^2 - 30*x*e^x + 90*x + 9))
Time = 13.98 (sec) , antiderivative size = 1157, normalized size of antiderivative = 38.57 \begin {dmath*} \int \frac {e^{\frac {2 \left (-198 x-558 x^2-18 x^3+288 x^4-72 x^5+e^{2 x} \left (-50 x^2-10 x^3+32 x^4-8 x^5\right )+e^x \left (60 x+336 x^2+36 x^3-192 x^4+48 x^5\right )\right )}{9+90 x+189 x^2-180 x^3+36 x^4+e^x \left (-30 x-138 x^2+120 x^3-24 x^4\right )+e^{2 x} \left (25 x^2-20 x^3+4 x^4\right )}} \left (-1188-756 x-7452 x^2-7020 x^3+14904 x^4-6480 x^5+864 x^6+e^x \left (360+612 x+5040 x^2+9180 x^3-15336 x^4+6480 x^5-864 x^6\right )+e^{3 x} \left (500 x^3-600 x^4+240 x^5-32 x^6\right )+e^{2 x} \left (-900 x^2-3780 x^3+5256 x^4-2160 x^5+288 x^6\right )\right )}{27+405 x+1863 x^2+1755 x^3-3726 x^4+1620 x^5-216 x^6+e^{2 x} \left (225 x^2+945 x^3-1314 x^4+540 x^5-72 x^6\right )+e^{3 x} \left (-125 x^3+150 x^4-60 x^5+8 x^6\right )+e^x \left (-135 x-1296 x^2-2295 x^3+3834 x^4-1620 x^5+216 x^6\right )} \, dx=\text {Too large to display} \end {dmath*}
int(-(exp(-(2*(198*x - exp(x)*(60*x + 336*x^2 + 36*x^3 - 192*x^4 + 48*x^5) + exp(2*x)*(50*x^2 + 10*x^3 - 32*x^4 + 8*x^5) + 558*x^2 + 18*x^3 - 288*x^ 4 + 72*x^5))/(90*x - exp(x)*(30*x + 138*x^2 - 120*x^3 + 24*x^4) + exp(2*x) *(25*x^2 - 20*x^3 + 4*x^4) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*(756*x + exp (2*x)*(900*x^2 + 3780*x^3 - 5256*x^4 + 2160*x^5 - 288*x^6) - exp(x)*(612*x + 5040*x^2 + 9180*x^3 - 15336*x^4 + 6480*x^5 - 864*x^6 + 360) - exp(3*x)* (500*x^3 - 600*x^4 + 240*x^5 - 32*x^6) + 7452*x^2 + 7020*x^3 - 14904*x^4 + 6480*x^5 - 864*x^6 + 1188))/(405*x + exp(2*x)*(225*x^2 + 945*x^3 - 1314*x ^4 + 540*x^5 - 72*x^6) - exp(x)*(135*x + 1296*x^2 + 2295*x^3 - 3834*x^4 + 1620*x^5 - 216*x^6) - exp(3*x)*(125*x^3 - 150*x^4 + 60*x^5 - 8*x^6) + 1863 *x^2 + 1755*x^3 - 3726*x^4 + 1620*x^5 - 216*x^6 + 27),x)
exp((72*x^3*exp(x))/(90*x - 138*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x ) + 25*x^2*exp(2*x) - 20*x^3*exp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189 *x^2 - 180*x^3 + 36*x^4 + 9))*exp((96*x^5*exp(x))/(90*x - 138*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x) + 25*x^2*exp(2*x) - 20*x^3*exp(2*x) + 4*x^ 4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*exp(-(384*x^4* exp(x))/(90*x - 138*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x) + 25*x^2*e xp(2*x) - 20*x^3*exp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x ^3 + 36*x^4 + 9))*exp((672*x^2*exp(x))/(90*x - 138*x^2*exp(x) + 120*x^3*ex p(x) - 24*x^4*exp(x) + 25*x^2*exp(2*x) - 20*x^3*exp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*exp(-(36*x^3)/(90*x - 138 *x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x) + 25*x^2*exp(2*x) - 20*x^3*ex p(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*e xp(-(144*x^5)/(90*x - 138*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x) + 25 *x^2*exp(2*x) - 20*x^3*exp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*exp((576*x^4)/(90*x - 138*x^2*exp(x) + 120*x^3*exp (x) - 24*x^4*exp(x) + 25*x^2*exp(2*x) - 20*x^3*exp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))*exp(-(1116*x^2)/(90*x - 13 8*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4*exp(x) + 25*x^2*exp(2*x) - 20*x^3*e xp(2*x) + 4*x^4*exp(2*x) - 30*x*exp(x) + 189*x^2 - 180*x^3 + 36*x^4 + 9))* exp(-(16*x^5*exp(2*x))/(90*x - 138*x^2*exp(x) + 120*x^3*exp(x) - 24*x^4...