Integrand size = 298, antiderivative size = 34 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=\left (2+x-e^{2 x-\frac {x}{4 \left (5+e^{(3-x) x^2}\right )}} x\right )^2 \] Output:
(x+2-x*exp(2*x-1/4*x/(exp((3-x)*x^2)+5)))^2
\[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=\int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx \] Input:
Integrate[(200 + 100*x + E^(6*x^2 - 2*x^3)*(8 + 4*x) + E^(3*x^2 - x^3)*(80 + 40*x) + E^((2*(39*x + 8*E^(3*x^2 - x^3)*x))/(20 + 4*E^(3*x^2 - x^3)))*( 100*x + 195*x^2 + E^(6*x^2 - 2*x^3)*(4*x + 8*x^2) + E^(3*x^2 - x^3)*(40*x + 79*x^2 + 6*x^4 - 3*x^5)) + E^((39*x + 8*E^(3*x^2 - x^3)*x)/(20 + 4*E^(3* x^2 - x^3)))*(-200 - 590*x - 195*x^2 + E^(6*x^2 - 2*x^3)*(-8 - 24*x - 8*x^ 2) + E^(3*x^2 - x^3)*(-80 - 238*x - 79*x^2 - 12*x^3 + 3*x^5)))/(50 + 2*E^( 6*x^2 - 2*x^3) + 20*E^(3*x^2 - x^3)),x]
Output:
Integrate[(200 + 100*x + E^(6*x^2 - 2*x^3)*(8 + 4*x) + E^(3*x^2 - x^3)*(80 + 40*x) + E^((2*(39*x + 8*E^(3*x^2 - x^3)*x))/(20 + 4*E^(3*x^2 - x^3)))*( 100*x + 195*x^2 + E^(6*x^2 - 2*x^3)*(4*x + 8*x^2) + E^(3*x^2 - x^3)*(40*x + 79*x^2 + 6*x^4 - 3*x^5)) + E^((39*x + 8*E^(3*x^2 - x^3)*x)/(20 + 4*E^(3* x^2 - x^3)))*(-200 - 590*x - 195*x^2 + E^(6*x^2 - 2*x^3)*(-8 - 24*x - 8*x^ 2) + E^(3*x^2 - x^3)*(-80 - 238*x - 79*x^2 - 12*x^3 + 3*x^5)))/(50 + 2*E^( 6*x^2 - 2*x^3) + 20*E^(3*x^2 - x^3)), 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.
| \(\displaystyle \int \frac {\left (-195 x^2+e^{6 x^2-2 x^3} \left (-8 x^2-24 x-8\right )+e^{3 x^2-x^3} \left (3 x^5-12 x^3-79 x^2-238 x-80\right )-590 x-200\right ) \exp \left (\frac {8 e^{3 x^2-x^3} x+39 x}{4 e^{3 x^2-x^3}+20}\right )+\left (195 x^2+e^{6 x^2-2 x^3} \left (8 x^2+4 x\right )+e^{3 x^2-x^3} \left (-3 x^5+6 x^4+79 x^2+40 x\right )+100 x\right ) \exp \left (\frac {2 \left (8 e^{3 x^2-x^3} x+39 x\right )}{4 e^{3 x^2-x^3}+20}\right )+e^{6 x^2-2 x^3} (4 x+8)+e^{3 x^2-x^3} (40 x+80)+100 x+200}{2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}+50} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x^3} \left (\left (-195 x^2+e^{6 x^2-2 x^3} \left (-8 x^2-24 x-8\right )+e^{3 x^2-x^3} \left (3 x^5-12 x^3-79 x^2-238 x-80\right )-590 x-200\right ) \exp \left (\frac {8 e^{3 x^2-x^3} x+39 x}{4 e^{3 x^2-x^3}+20}\right )+\left (195 x^2+e^{6 x^2-2 x^3} \left (8 x^2+4 x\right )+e^{3 x^2-x^3} \left (-3 x^5+6 x^4+79 x^2+40 x\right )+100 x\right ) \exp \left (\frac {2 \left (8 e^{3 x^2-x^3} x+39 x\right )}{4 e^{3 x^2-x^3}+20}\right )+e^{6 x^2-2 x^3} (4 x+8)+e^{3 x^2-x^3} (40 x+80)+100 x+200\right )}{2 \left (5 e^{x^3}+e^{3 x^2}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{2 x^3} \left (100 x+4 e^{6 x^2-2 x^3} (x+2)+40 e^{3 x^2-x^3} (x+2)-\exp \left (\frac {8 e^{3 x^2-x^3} x+39 x}{4 \left (5+e^{3 x^2-x^3}\right )}\right ) \left (195 x^2+590 x+8 e^{6 x^2-2 x^3} \left (x^2+3 x+1\right )+e^{3 x^2-x^3} \left (-3 x^5+12 x^3+79 x^2+238 x+80\right )+200\right )+\exp \left (\frac {8 e^{3 x^2-x^3} x+39 x}{2 \left (5+e^{3 x^2-x^3}\right )}\right ) \left (195 x^2+100 x+4 e^{6 x^2-2 x^3} \left (2 x^2+x\right )+e^{3 x^2-x^3} \left (-3 x^5+6 x^4+79 x^2+40 x\right )\right )+200\right )}{\left (e^{3 x^2}+5 e^{x^3}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {15 \exp \left (2 x^3+\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) (x-2) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) x^3}{\left (e^{3 x^2}+5 e^{x^3}\right )^2}-\exp \left (2 x^3-\frac {x \left (4 e^{3 x^2} \left (x^2+3 x-2\right )+e^{x^3} \left (20 x^2+60 x-39\right )\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x+\frac {5 \exp \left (2 x^3-\frac {x \left (4 e^{3 x^2} (3 x-2)+e^{x^3} (60 x-39)\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x}{e^{3 x^2}+5 e^{x^3}}+4 \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (2 \exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x+\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right )-1\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {15 \exp \left (2 x^3+\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) (x-2) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) x^3}{\left (e^{3 x^2}+5 e^{x^3}\right )^2}-\exp \left (\frac {x \left (4 e^{3 x^2} \left (x^2-3 x+2\right )+e^{x^3} \left (20 x^2-60 x+39\right )\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x+\frac {5 \exp \left (2 x^3-\frac {x \left (4 e^{3 x^2} (3 x-2)+e^{x^3} (60 x-39)\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x}{e^{3 x^2}+5 e^{x^3}}+4 \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (2 \exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x+\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right )-1\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {15 \exp \left (2 x^3+\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) (x-2) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) x^3}{\left (e^{3 x^2}+5 e^{x^3}\right )^2}-\exp \left (\frac {x \left (4 e^{3 x^2} \left (x^2-3 x+2\right )+e^{x^3} \left (20 x^2-60 x+39\right )\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x+\frac {5 \exp \left (2 x^3-\frac {x \left (4 e^{3 x^2} (3 x-2)+e^{x^3} (60 x-39)\right )}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (3 x^3-6 x^2+1\right ) x}{e^{3 x^2}+5 e^{x^3}}+4 \left (\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x-x-2\right ) \left (2 \exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right ) x+\exp \left (\frac {\left (8 e^{3 x^2}+39 e^{x^3}\right ) x}{4 \left (e^{3 x^2}+5 e^{x^3}\right )}\right )-1\right )\right )dx\) |
Input:
Int[(200 + 100*x + E^(6*x^2 - 2*x^3)*(8 + 4*x) + E^(3*x^2 - x^3)*(80 + 40* x) + E^((2*(39*x + 8*E^(3*x^2 - x^3)*x))/(20 + 4*E^(3*x^2 - x^3)))*(100*x + 195*x^2 + E^(6*x^2 - 2*x^3)*(4*x + 8*x^2) + E^(3*x^2 - x^3)*(40*x + 79*x ^2 + 6*x^4 - 3*x^5)) + E^((39*x + 8*E^(3*x^2 - x^3)*x)/(20 + 4*E^(3*x^2 - x^3)))*(-200 - 590*x - 195*x^2 + E^(6*x^2 - 2*x^3)*(-8 - 24*x - 8*x^2) + E ^(3*x^2 - x^3)*(-80 - 238*x - 79*x^2 - 12*x^3 + 3*x^5)))/(50 + 2*E^(6*x^2 - 2*x^3) + 20*E^(3*x^2 - x^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(30)=60\).
Time = 2.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.41
| method | result | size |
| risch | \(x^{2} {\mathrm e}^{\frac {\left (8 \,{\mathrm e}^{-x^{2} \left (-3+x \right )}+39\right ) x}{2 \,{\mathrm e}^{-x^{2} \left (-3+x \right )}+10}}+x^{2}+4 x +\left (-2 x^{2}-4 x \right ) {\mathrm e}^{\frac {\left (8 \,{\mathrm e}^{-x^{2} \left (-3+x \right )}+39\right ) x}{4 \,{\mathrm e}^{-x^{2} \left (-3+x \right )}+20}}\) | \(82\) |
| parallelrisch | \({\mathrm e}^{\frac {x \left (8 \,{\mathrm e}^{-x^{3}+3 x^{2}}+39\right )}{2 \,{\mathrm e}^{-x^{3}+3 x^{2}}+10}} x^{2}-2 \,{\mathrm e}^{\frac {x \left (8 \,{\mathrm e}^{-x^{3}+3 x^{2}}+39\right )}{4 \,{\mathrm e}^{-x^{3}+3 x^{2}}+20}} x^{2}+x^{2}-4 \,{\mathrm e}^{\frac {x \left (8 \,{\mathrm e}^{-x^{3}+3 x^{2}}+39\right )}{4 \,{\mathrm e}^{-x^{3}+3 x^{2}}+20}} x +4 x\) | \(130\) |
Input:
int((((8*x^2+4*x)*exp(-x^3+3*x^2)^2+(-3*x^5+6*x^4+79*x^2+40*x)*exp(-x^3+3* x^2)+195*x^2+100*x)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3*x^2)+20)) ^2+((-8*x^2-24*x-8)*exp(-x^3+3*x^2)^2+(3*x^5-12*x^3-79*x^2-238*x-80)*exp(- x^3+3*x^2)-195*x^2-590*x-200)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3 *x^2)+20))+(4*x+8)*exp(-x^3+3*x^2)^2+(40*x+80)*exp(-x^3+3*x^2)+100*x+200)/ (2*exp(-x^3+3*x^2)^2+20*exp(-x^3+3*x^2)+50),x,method=_RETURNVERBOSE)
Output:
x^2*exp(1/2*x*(8*exp(-x^2*(-3+x))+39)/(exp(-x^2*(-3+x))+5))+x^2+4*x+(-2*x^ 2-4*x)*exp(1/4*x*(8*exp(-x^2*(-3+x))+39)/(exp(-x^2*(-3+x))+5))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.82 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=x^{2} e^{\left (\frac {8 \, x e^{\left (-x^{3} + 3 \, x^{2}\right )} + 39 \, x}{2 \, {\left (e^{\left (-x^{3} + 3 \, x^{2}\right )} + 5\right )}}\right )} + x^{2} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {8 \, x e^{\left (-x^{3} + 3 \, x^{2}\right )} + 39 \, x}{4 \, {\left (e^{\left (-x^{3} + 3 \, x^{2}\right )} + 5\right )}}\right )} + 4 \, x \] Input:
integrate((((8*x^2+4*x)*exp(-x^3+3*x^2)^2+(-3*x^5+6*x^4+79*x^2+40*x)*exp(- x^3+3*x^2)+195*x^2+100*x)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3*x^2 )+20))^2+((-8*x^2-24*x-8)*exp(-x^3+3*x^2)^2+(3*x^5-12*x^3-79*x^2-238*x-80) *exp(-x^3+3*x^2)-195*x^2-590*x-200)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp( -x^3+3*x^2)+20))+(4*x+8)*exp(-x^3+3*x^2)^2+(40*x+80)*exp(-x^3+3*x^2)+100*x +200)/(2*exp(-x^3+3*x^2)^2+20*exp(-x^3+3*x^2)+50),x, algorithm="fricas")
Output:
x^2*e^(1/2*(8*x*e^(-x^3 + 3*x^2) + 39*x)/(e^(-x^3 + 3*x^2) + 5)) + x^2 - 2 *(x^2 + 2*x)*e^(1/4*(8*x*e^(-x^3 + 3*x^2) + 39*x)/(e^(-x^3 + 3*x^2) + 5)) + 4*x
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).
Time = 24.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=x^{2} e^{\frac {2 \cdot \left (8 x e^{- x^{3} + 3 x^{2}} + 39 x\right )}{4 e^{- x^{3} + 3 x^{2}} + 20}} + x^{2} + 4 x + \left (- 2 x^{2} - 4 x\right ) e^{\frac {8 x e^{- x^{3} + 3 x^{2}} + 39 x}{4 e^{- x^{3} + 3 x^{2}} + 20}} \] Input:
integrate((((8*x**2+4*x)*exp(-x**3+3*x**2)**2+(-3*x**5+6*x**4+79*x**2+40*x )*exp(-x**3+3*x**2)+195*x**2+100*x)*exp((8*x*exp(-x**3+3*x**2)+39*x)/(4*ex p(-x**3+3*x**2)+20))**2+((-8*x**2-24*x-8)*exp(-x**3+3*x**2)**2+(3*x**5-12* x**3-79*x**2-238*x-80)*exp(-x**3+3*x**2)-195*x**2-590*x-200)*exp((8*x*exp( -x**3+3*x**2)+39*x)/(4*exp(-x**3+3*x**2)+20))+(4*x+8)*exp(-x**3+3*x**2)**2 +(40*x+80)*exp(-x**3+3*x**2)+100*x+200)/(2*exp(-x**3+3*x**2)**2+20*exp(-x* *3+3*x**2)+50),x)
Output:
x**2*exp(2*(8*x*exp(-x**3 + 3*x**2) + 39*x)/(4*exp(-x**3 + 3*x**2) + 20)) + x**2 + 4*x + (-2*x**2 - 4*x)*exp((8*x*exp(-x**3 + 3*x**2) + 39*x)/(4*exp (-x**3 + 3*x**2) + 20))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=x^{2} e^{\left (\frac {39 \, x e^{\left (x^{3}\right )}}{2 \, {\left (5 \, e^{\left (x^{3}\right )} + e^{\left (3 \, x^{2}\right )}\right )}} + \frac {4 \, x e^{\left (3 \, x^{2}\right )}}{5 \, e^{\left (x^{3}\right )} + e^{\left (3 \, x^{2}\right )}}\right )} + x^{2} - 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {39 \, x e^{\left (x^{3}\right )}}{4 \, {\left (5 \, e^{\left (x^{3}\right )} + e^{\left (3 \, x^{2}\right )}\right )}} + \frac {2 \, x e^{\left (3 \, x^{2}\right )}}{5 \, e^{\left (x^{3}\right )} + e^{\left (3 \, x^{2}\right )}}\right )} + 4 \, x \] Input:
integrate((((8*x^2+4*x)*exp(-x^3+3*x^2)^2+(-3*x^5+6*x^4+79*x^2+40*x)*exp(- x^3+3*x^2)+195*x^2+100*x)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3*x^2 )+20))^2+((-8*x^2-24*x-8)*exp(-x^3+3*x^2)^2+(3*x^5-12*x^3-79*x^2-238*x-80) *exp(-x^3+3*x^2)-195*x^2-590*x-200)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp( -x^3+3*x^2)+20))+(4*x+8)*exp(-x^3+3*x^2)^2+(40*x+80)*exp(-x^3+3*x^2)+100*x +200)/(2*exp(-x^3+3*x^2)^2+20*exp(-x^3+3*x^2)+50),x, algorithm="maxima")
Output:
x^2*e^(39/2*x*e^(x^3)/(5*e^(x^3) + e^(3*x^2)) + 4*x*e^(3*x^2)/(5*e^(x^3) + e^(3*x^2))) + x^2 - 2*(x^2 + 2*x)*e^(39/4*x*e^(x^3)/(5*e^(x^3) + e^(3*x^2 )) + 2*x*e^(3*x^2)/(5*e^(x^3) + e^(3*x^2))) + 4*x
Timed out. \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=\text {Timed out} \] Input:
integrate((((8*x^2+4*x)*exp(-x^3+3*x^2)^2+(-3*x^5+6*x^4+79*x^2+40*x)*exp(- x^3+3*x^2)+195*x^2+100*x)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3*x^2 )+20))^2+((-8*x^2-24*x-8)*exp(-x^3+3*x^2)^2+(3*x^5-12*x^3-79*x^2-238*x-80) *exp(-x^3+3*x^2)-195*x^2-590*x-200)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp( -x^3+3*x^2)+20))+(4*x+8)*exp(-x^3+3*x^2)^2+(40*x+80)*exp(-x^3+3*x^2)+100*x +200)/(2*exp(-x^3+3*x^2)^2+20*exp(-x^3+3*x^2)+50),x, algorithm="giac")
Output:
Timed out
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.94 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=4\,x-{\mathrm {e}}^{\frac {39\,x}{4\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}+20}+\frac {8\,x\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}}{4\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}+20}}\,\left (2\,x^2+4\,x\right )+x^2\,{\mathrm {e}}^{\frac {78\,x}{4\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}+20}+\frac {16\,x\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}}{4\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{3\,x^2}+20}}+x^2 \] Input:
int((100*x + exp((2*(39*x + 8*x*exp(3*x^2 - x^3)))/(4*exp(3*x^2 - x^3) + 2 0))*(100*x + exp(3*x^2 - x^3)*(40*x + 79*x^2 + 6*x^4 - 3*x^5) + 195*x^2 + exp(6*x^2 - 2*x^3)*(4*x + 8*x^2)) - exp((39*x + 8*x*exp(3*x^2 - x^3))/(4*e xp(3*x^2 - x^3) + 20))*(590*x + exp(3*x^2 - x^3)*(238*x + 79*x^2 + 12*x^3 - 3*x^5 + 80) + 195*x^2 + exp(6*x^2 - 2*x^3)*(24*x + 8*x^2 + 8) + 200) + e xp(6*x^2 - 2*x^3)*(4*x + 8) + exp(3*x^2 - x^3)*(40*x + 80) + 200)/(20*exp( 3*x^2 - x^3) + 2*exp(6*x^2 - 2*x^3) + 50),x)
Output:
4*x - exp((39*x)/(4*exp(-x^3)*exp(3*x^2) + 20) + (8*x*exp(-x^3)*exp(3*x^2) )/(4*exp(-x^3)*exp(3*x^2) + 20))*(4*x + 2*x^2) + x^2*exp((78*x)/(4*exp(-x^ 3)*exp(3*x^2) + 20) + (16*x*exp(-x^3)*exp(3*x^2))/(4*exp(-x^3)*exp(3*x^2) + 20)) + x^2
Time = 1.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 3.97 \[ \int \frac {200+100 x+e^{6 x^2-2 x^3} (8+4 x)+e^{3 x^2-x^3} (80+40 x)+e^{\frac {2 \left (39 x+8 e^{3 x^2-x^3} x\right )}{20+4 e^{3 x^2-x^3}}} \left (100 x+195 x^2+e^{6 x^2-2 x^3} \left (4 x+8 x^2\right )+e^{3 x^2-x^3} \left (40 x+79 x^2+6 x^4-3 x^5\right )\right )+e^{\frac {39 x+8 e^{3 x^2-x^3} x}{20+4 e^{3 x^2-x^3}}} \left (-200-590 x-195 x^2+e^{6 x^2-2 x^3} \left (-8-24 x-8 x^2\right )+e^{3 x^2-x^3} \left (-80-238 x-79 x^2-12 x^3+3 x^5\right )\right )}{50+2 e^{6 x^2-2 x^3}+20 e^{3 x^2-x^3}} \, dx=x \left (-2 e^{\frac {39 e^{x^{3}} x +8 e^{3 x^{2}} x}{20 e^{x^{3}}+4 e^{3 x^{2}}}} x -4 e^{\frac {39 e^{x^{3}} x +8 e^{3 x^{2}} x}{20 e^{x^{3}}+4 e^{3 x^{2}}}}+e^{\frac {39 e^{x^{3}} x +8 e^{3 x^{2}} x}{10 e^{x^{3}}+2 e^{3 x^{2}}}} x +x +4\right ) \] Input:
int((((8*x^2+4*x)*exp(-x^3+3*x^2)^2+(-3*x^5+6*x^4+79*x^2+40*x)*exp(-x^3+3* x^2)+195*x^2+100*x)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3*x^2)+20)) ^2+((-8*x^2-24*x-8)*exp(-x^3+3*x^2)^2+(3*x^5-12*x^3-79*x^2-238*x-80)*exp(- x^3+3*x^2)-195*x^2-590*x-200)*exp((8*x*exp(-x^3+3*x^2)+39*x)/(4*exp(-x^3+3 *x^2)+20))+(4*x+8)*exp(-x^3+3*x^2)^2+(40*x+80)*exp(-x^3+3*x^2)+100*x+200)/ (2*exp(-x^3+3*x^2)^2+20*exp(-x^3+3*x^2)+50),x)
Output:
x*( - 2*e**((39*e**(x**3)*x + 8*e**(3*x**2)*x)/(20*e**(x**3) + 4*e**(3*x** 2)))*x - 4*e**((39*e**(x**3)*x + 8*e**(3*x**2)*x)/(20*e**(x**3) + 4*e**(3* x**2))) + e**((39*e**(x**3)*x + 8*e**(3*x**2)*x)/(10*e**(x**3) + 2*e**(3*x **2)))*x + x + 4)