Integrand size = 289, antiderivative size = 27 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=3+x+\left (2+e^{20}+x-\frac {5}{-e^4+e^{x^2}+x}\right )^2 \] Output:
3+x+(exp(20)+2-5/(exp(x^2)-exp(4)+x)+x)^2
Time = 10.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\left (5+2 e^{20}\right ) x+x^2+\frac {25}{\left (-e^4+e^{x^2}+x\right )^2}-\frac {10 \left (2+e^{20}+x\right )}{-e^4+e^{x^2}+x} \] Input:
Integrate[(-50 + E^12*(-5 - 2*x) + 20*x + 5*x^3 + 2*x^4 + E^(3*x^2)*(5 + 2 *E^20 + 2*x) + E^8*(-10 + 15*x + 6*x^2) + E^4*(-20 + 10*x - 15*x^2 - 6*x^3 ) + E^(2*x^2)*(-10 + E^4*(-15 - 6*x) + 55*x + 26*x^2 + E^20*(-6*E^4 + 26*x )) + E^20*(-2*E^12 + 10*x + 6*E^8*x + 2*x^3 + E^4*(-10 - 6*x^2)) + E^x^2*( 20 - 110*x + 55*x^2 + 26*x^3 + E^8*(15 + 6*x) + E^4*(20 - 70*x - 32*x^2) + E^20*(10 + 6*E^8 - 32*E^4*x + 26*x^2)))/(-E^12 + E^(3*x^2) + 3*E^8*x - 3* E^4*x^2 + x^3 + E^(2*x^2)*(-3*E^4 + 3*x) + E^x^2*(3*E^8 - 6*E^4*x + 3*x^2) ),x]
Output:
(5 + 2*E^20)*x + x^2 + 25/(-E^4 + E^x^2 + x)^2 - (10*(2 + E^20 + x))/(-E^4 + E^x^2 + x)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 x^4+5 x^3+e^{3 x^2} \left (2 x+2 e^{20}+5\right )+e^8 \left (6 x^2+15 x-10\right )+e^{2 x^2} \left (26 x^2+55 x+e^4 (-6 x-15)+e^{20} \left (26 x-6 e^4\right )-10\right )+e^4 \left (-6 x^3-15 x^2+10 x-20\right )+e^{20} \left (2 x^3+e^4 \left (-6 x^2-10\right )+6 e^8 x+10 x-2 e^{12}\right )+e^{x^2} \left (26 x^3+55 x^2+e^4 \left (-32 x^2-70 x+20\right )+e^{20} \left (26 x^2-32 e^4 x+6 e^8+10\right )-110 x+e^8 (6 x+15)+20\right )+20 x+e^{12} (-2 x-5)-50}{x^3-3 e^4 x^2+e^{3 x^2}+e^{2 x^2} \left (3 x-3 e^4\right )+e^{x^2} \left (3 x^2-6 e^4 x+3 e^8\right )+3 e^8 x-e^{12}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^4-5 x^3-e^{3 x^2} \left (2 x+2 e^{20}+5\right )-e^8 \left (6 x^2+15 x-10\right )-e^{2 x^2} \left (26 x^2+55 x+e^4 (-6 x-15)+e^{20} \left (26 x-6 e^4\right )-10\right )-e^4 \left (-6 x^3-15 x^2+10 x-20\right )-e^{20} \left (2 x^3+e^4 \left (-6 x^2-10\right )+6 e^8 x+10 x-2 e^{12}\right )-e^{x^2} \left (26 x^3+55 x^2+e^4 \left (-32 x^2-70 x+20\right )+e^{20} \left (26 x^2-32 e^4 x+6 e^8+10\right )-110 x+e^8 (6 x+15)+20\right )-20 x-e^{12} (-2 x-5)+50}{\left (-e^{x^2}-x+e^4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {50 \left (-2 x^2+2 e^4 x+1\right )}{\left (e^{x^2}+x-e^4\right )^3}+\frac {10 \left (-2 x^2-2 \left (2+e^{20}\right ) x+1\right )}{-e^{x^2}-x+e^4}+\frac {10 \left (-2 x^3-2 \left (2-e^4+e^{20}\right ) x^2-\left (9-4 e^4-2 e^{24}\right ) x+e^{20}+2\right )}{\left (-e^{x^2}-x+e^4\right )^2}+2 x+5 \left (1+\frac {2 e^{20}}{5}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 10 \left (2+e^{20}\right ) \int \frac {1}{\left (-x-e^{x^2}+e^4\right )^2}dx+100 e^4 \int \frac {x}{\left (-x-e^{x^2}+e^4\right )^3}dx-20 \left (2+e^{20}\right ) \int \frac {x}{-x-e^{x^2}+e^4}dx-20 \left (2-e^4+e^{20}\right ) \int \frac {x^2}{\left (-x-e^{x^2}+e^4\right )^2}dx-50 \int \frac {1}{\left (x+e^{x^2}-e^4\right )^3}dx+100 \int \frac {x^2}{\left (x+e^{x^2}-e^4\right )^3}dx-10 \left (9-4 e^4-2 e^{24}\right ) \int \frac {x}{\left (x+e^{x^2}-e^4\right )^2}dx-10 \int \frac {1}{x+e^{x^2}-e^4}dx+20 \int \frac {x^2}{x+e^{x^2}-e^4}dx-20 \int \frac {x^3}{\left (x+e^{x^2}-e^4\right )^2}dx+x^2+\left (5+2 e^{20}\right ) x\) |
Input:
Int[(-50 + E^12*(-5 - 2*x) + 20*x + 5*x^3 + 2*x^4 + E^(3*x^2)*(5 + 2*E^20 + 2*x) + E^8*(-10 + 15*x + 6*x^2) + E^4*(-20 + 10*x - 15*x^2 - 6*x^3) + E^ (2*x^2)*(-10 + E^4*(-15 - 6*x) + 55*x + 26*x^2 + E^20*(-6*E^4 + 26*x)) + E ^20*(-2*E^12 + 10*x + 6*E^8*x + 2*x^3 + E^4*(-10 - 6*x^2)) + E^x^2*(20 - 1 10*x + 55*x^2 + 26*x^3 + E^8*(15 + 6*x) + E^4*(20 - 70*x - 32*x^2) + E^20* (10 + 6*E^8 - 32*E^4*x + 26*x^2)))/(-E^12 + E^(3*x^2) + 3*E^8*x - 3*E^4*x^ 2 + x^3 + E^(2*x^2)*(-3*E^4 + 3*x) + E^x^2*(3*E^8 - 6*E^4*x + 3*x^2)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(24)=48\).
Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 8.81
\[\frac {x^{4}+x^{2} {\mathrm e}^{2 x^{2}}+\left (-2 \,{\mathrm e}^{4}+5+2 \,{\mathrm e}^{20}\right ) x^{3}+\left (4 \,{\mathrm e}^{20} {\mathrm e}^{4}-{\mathrm e}^{8}+10 \,{\mathrm e}^{4}+10\right ) {\mathrm e}^{2 x^{2}}+\left (-8 \,{\mathrm e}^{20} {\mathrm e}^{8}+2 \,{\mathrm e}^{12}-20 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{20}-20 \,{\mathrm e}^{4}-20\right ) {\mathrm e}^{x^{2}}+\left (-6 \,{\mathrm e}^{20} {\mathrm e}^{8}+2 \,{\mathrm e}^{12}-15 \,{\mathrm e}^{8}-10 \,{\mathrm e}^{20}-10 \,{\mathrm e}^{4}-20\right ) x +\left (2 \,{\mathrm e}^{20}+5\right ) x \,{\mathrm e}^{2 x^{2}}+\left (-2 \,{\mathrm e}^{4}+10+4 \,{\mathrm e}^{20}\right ) x^{2} {\mathrm e}^{x^{2}}+\left (4 \,{\mathrm e}^{20} {\mathrm e}^{4}-2 \,{\mathrm e}^{8}+10 \,{\mathrm e}^{4}+10\right ) x \,{\mathrm e}^{x^{2}}+2 x^{3} {\mathrm e}^{x^{2}}+25+4 \,{\mathrm e}^{12} {\mathrm e}^{20}-{\mathrm e}^{16}+10 \,{\mathrm e}^{12}+10 \,{\mathrm e}^{20} {\mathrm e}^{4}+10 \,{\mathrm e}^{8}+20 \,{\mathrm e}^{4}}{\left ({\mathrm e}^{4}-x -{\mathrm e}^{x^{2}}\right )^{2}}\]
Input:
int(((2*exp(20)+5+2*x)*exp(x^2)^3+((-6*exp(4)+26*x)*exp(20)+(-6*x-15)*exp( 4)+26*x^2+55*x-10)*exp(x^2)^2+((6*exp(4)^2-32*x*exp(4)+26*x^2+10)*exp(20)+ (6*x+15)*exp(4)^2+(-32*x^2-70*x+20)*exp(4)+26*x^3+55*x^2-110*x+20)*exp(x^2 )+(-2*exp(4)^3+6*x*exp(4)^2+(-6*x^2-10)*exp(4)+2*x^3+10*x)*exp(20)+(-2*x-5 )*exp(4)^3+(6*x^2+15*x-10)*exp(4)^2+(-6*x^3-15*x^2+10*x-20)*exp(4)+2*x^4+5 *x^3+20*x-50)/(exp(x^2)^3+(-3*exp(4)+3*x)*exp(x^2)^2+(3*exp(4)^2-6*x*exp(4 )+3*x^2)*exp(x^2)-exp(4)^3+3*x*exp(4)^2-3*x^2*exp(4)+x^3),x)
Output:
(x^4+x^2*exp(x^2)^2+(-2*exp(4)+5+2*exp(20))*x^3+(4*exp(20)*exp(4)-exp(4)^2 +10*exp(4)+10)*exp(x^2)^2+(-8*exp(20)*exp(4)^2+2*exp(4)^3-20*exp(4)^2-10*e xp(20)-20*exp(4)-20)*exp(x^2)+(-6*exp(20)*exp(4)^2+2*exp(4)^3-15*exp(4)^2- 10*exp(20)-10*exp(4)-20)*x+(2*exp(20)+5)*x*exp(x^2)^2+(-2*exp(4)+10+4*exp( 20))*x^2*exp(x^2)+(4*exp(20)*exp(4)-2*exp(4)^2+10*exp(4)+10)*x*exp(x^2)+2* x^3*exp(x^2)+25+4*exp(20)*exp(4)^3-exp(4)^4+10*exp(4)^3+10*exp(20)*exp(4)+ 10*exp(4)^2+20*exp(4))/(exp(4)-x-exp(x^2))^2
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.22 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {x^{4} + 5 \, x^{3} - 10 \, x^{2} + 2 \, x e^{28} - 2 \, {\left (2 \, x^{2} - 5\right )} e^{24} + 2 \, {\left (x^{3} - 5 \, x\right )} e^{20} + {\left (x^{2} + 5 \, x\right )} e^{8} - 2 \, {\left (x^{3} + 5 \, x^{2} - 5 \, x - 10\right )} e^{4} + {\left (x^{2} + 2 \, x e^{20} + 5 \, x\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + 5 \, x^{2} - 2 \, x e^{24} + {\left (2 \, x^{2} - 5\right )} e^{20} - {\left (x^{2} + 5 \, x\right )} e^{4} - 5 \, x - 10\right )} e^{\left (x^{2}\right )} - 20 \, x + 25}{x^{2} - 2 \, x e^{4} + 2 \, {\left (x - e^{4}\right )} e^{\left (x^{2}\right )} + e^{8} + e^{\left (2 \, x^{2}\right )}} \] Input:
integrate(((2*exp(20)+5+2*x)*exp(x^2)^3+((-6*exp(4)+26*x)*exp(20)+(-6*x-15 )*exp(4)+26*x^2+55*x-10)*exp(x^2)^2+((6*exp(4)^2-32*x*exp(4)+26*x^2+10)*ex p(20)+(6*x+15)*exp(4)^2+(-32*x^2-70*x+20)*exp(4)+26*x^3+55*x^2-110*x+20)*e xp(x^2)+(-2*exp(4)^3+6*x*exp(4)^2+(-6*x^2-10)*exp(4)+2*x^3+10*x)*exp(20)+( -2*x-5)*exp(4)^3+(6*x^2+15*x-10)*exp(4)^2+(-6*x^3-15*x^2+10*x-20)*exp(4)+2 *x^4+5*x^3+20*x-50)/(exp(x^2)^3+(-3*exp(4)+3*x)*exp(x^2)^2+(3*exp(4)^2-6*x *exp(4)+3*x^2)*exp(x^2)-exp(4)^3+3*x*exp(4)^2-3*x^2*exp(4)+x^3),x, algorit hm="fricas")
Output:
(x^4 + 5*x^3 - 10*x^2 + 2*x*e^28 - 2*(2*x^2 - 5)*e^24 + 2*(x^3 - 5*x)*e^20 + (x^2 + 5*x)*e^8 - 2*(x^3 + 5*x^2 - 5*x - 10)*e^4 + (x^2 + 2*x*e^20 + 5* x)*e^(2*x^2) + 2*(x^3 + 5*x^2 - 2*x*e^24 + (2*x^2 - 5)*e^20 - (x^2 + 5*x)* e^4 - 5*x - 10)*e^(x^2) - 20*x + 25)/(x^2 - 2*x*e^4 + 2*(x - e^4)*e^(x^2) + e^8 + e^(2*x^2))
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=x^{2} + x \left (5 + 2 e^{20}\right ) + \frac {- 10 x^{2} - 10 x e^{20} - 20 x + 10 x e^{4} + \left (- 10 x - 10 e^{20} - 20\right ) e^{x^{2}} + 25 + 20 e^{4} + 10 e^{24}}{x^{2} - 2 x e^{4} + \left (2 x - 2 e^{4}\right ) e^{x^{2}} + e^{2 x^{2}} + e^{8}} \] Input:
integrate(((2*exp(20)+5+2*x)*exp(x**2)**3+((-6*exp(4)+26*x)*exp(20)+(-6*x- 15)*exp(4)+26*x**2+55*x-10)*exp(x**2)**2+((6*exp(4)**2-32*x*exp(4)+26*x**2 +10)*exp(20)+(6*x+15)*exp(4)**2+(-32*x**2-70*x+20)*exp(4)+26*x**3+55*x**2- 110*x+20)*exp(x**2)+(-2*exp(4)**3+6*x*exp(4)**2+(-6*x**2-10)*exp(4)+2*x**3 +10*x)*exp(20)+(-2*x-5)*exp(4)**3+(6*x**2+15*x-10)*exp(4)**2+(-6*x**3-15*x **2+10*x-20)*exp(4)+2*x**4+5*x**3+20*x-50)/(exp(x**2)**3+(-3*exp(4)+3*x)*e xp(x**2)**2+(3*exp(4)**2-6*x*exp(4)+3*x**2)*exp(x**2)-exp(4)**3+3*x*exp(4) **2-3*x**2*exp(4)+x**3),x)
Output:
x**2 + x*(5 + 2*exp(20)) + (-10*x**2 - 10*x*exp(20) - 20*x + 10*x*exp(4) + (-10*x - 10*exp(20) - 20)*exp(x**2) + 25 + 20*exp(4) + 10*exp(24))/(x**2 - 2*x*exp(4) + (2*x - 2*exp(4))*exp(x**2) + exp(2*x**2) + exp(8))
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 159, normalized size of antiderivative = 5.89 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {x^{4} + x^{3} {\left (2 \, e^{20} - 2 \, e^{4} + 5\right )} - x^{2} {\left (4 \, e^{24} - e^{8} + 10 \, e^{4} + 10\right )} + x {\left (2 \, e^{28} - 10 \, e^{20} + 5 \, e^{8} + 10 \, e^{4} - 20\right )} + {\left (x^{2} + x {\left (2 \, e^{20} + 5\right )}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} + x^{2} {\left (2 \, e^{20} - e^{4} + 5\right )} - x {\left (2 \, e^{24} + 5 \, e^{4} + 5\right )} - 5 \, e^{20} - 10\right )} e^{\left (x^{2}\right )} + 10 \, e^{24} + 20 \, e^{4} + 25}{x^{2} - 2 \, x e^{4} + 2 \, {\left (x - e^{4}\right )} e^{\left (x^{2}\right )} + e^{8} + e^{\left (2 \, x^{2}\right )}} \] Input:
integrate(((2*exp(20)+5+2*x)*exp(x^2)^3+((-6*exp(4)+26*x)*exp(20)+(-6*x-15 )*exp(4)+26*x^2+55*x-10)*exp(x^2)^2+((6*exp(4)^2-32*x*exp(4)+26*x^2+10)*ex p(20)+(6*x+15)*exp(4)^2+(-32*x^2-70*x+20)*exp(4)+26*x^3+55*x^2-110*x+20)*e xp(x^2)+(-2*exp(4)^3+6*x*exp(4)^2+(-6*x^2-10)*exp(4)+2*x^3+10*x)*exp(20)+( -2*x-5)*exp(4)^3+(6*x^2+15*x-10)*exp(4)^2+(-6*x^3-15*x^2+10*x-20)*exp(4)+2 *x^4+5*x^3+20*x-50)/(exp(x^2)^3+(-3*exp(4)+3*x)*exp(x^2)^2+(3*exp(4)^2-6*x *exp(4)+3*x^2)*exp(x^2)-exp(4)^3+3*x*exp(4)^2-3*x^2*exp(4)+x^3),x, algorit hm="maxima")
Output:
(x^4 + x^3*(2*e^20 - 2*e^4 + 5) - x^2*(4*e^24 - e^8 + 10*e^4 + 10) + x*(2* e^28 - 10*e^20 + 5*e^8 + 10*e^4 - 20) + (x^2 + x*(2*e^20 + 5))*e^(2*x^2) + 2*(x^3 + x^2*(2*e^20 - e^4 + 5) - x*(2*e^24 + 5*e^4 + 5) - 5*e^20 - 10)*e ^(x^2) + 10*e^24 + 20*e^4 + 25)/(x^2 - 2*x*e^4 + 2*(x - e^4)*e^(x^2) + e^8 + e^(2*x^2))
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 224, normalized size of antiderivative = 8.30 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {x^{4} + 2 \, x^{3} e^{20} - 2 \, x^{3} e^{4} + 2 \, x^{3} e^{\left (x^{2}\right )} + 5 \, x^{3} - 4 \, x^{2} e^{24} + x^{2} e^{8} - 10 \, x^{2} e^{4} + x^{2} e^{\left (2 \, x^{2}\right )} + 4 \, x^{2} e^{\left (x^{2} + 20\right )} - 2 \, x^{2} e^{\left (x^{2} + 4\right )} + 10 \, x^{2} e^{\left (x^{2}\right )} - 10 \, x^{2} + 2 \, x e^{28} - 10 \, x e^{20} + 5 \, x e^{8} + 10 \, x e^{4} + 5 \, x e^{\left (2 \, x^{2}\right )} + 2 \, x e^{\left (2 \, x^{2} + 20\right )} - 4 \, x e^{\left (x^{2} + 24\right )} - 10 \, x e^{\left (x^{2} + 4\right )} - 10 \, x e^{\left (x^{2}\right )} - 20 \, x + 10 \, e^{24} + 20 \, e^{4} - 10 \, e^{\left (x^{2} + 20\right )} - 20 \, e^{\left (x^{2}\right )} + 25}{x^{2} - 2 \, x e^{4} + 2 \, x e^{\left (x^{2}\right )} + e^{8} + e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2} + 4\right )}} \] Input:
integrate(((2*exp(20)+5+2*x)*exp(x^2)^3+((-6*exp(4)+26*x)*exp(20)+(-6*x-15 )*exp(4)+26*x^2+55*x-10)*exp(x^2)^2+((6*exp(4)^2-32*x*exp(4)+26*x^2+10)*ex p(20)+(6*x+15)*exp(4)^2+(-32*x^2-70*x+20)*exp(4)+26*x^3+55*x^2-110*x+20)*e xp(x^2)+(-2*exp(4)^3+6*x*exp(4)^2+(-6*x^2-10)*exp(4)+2*x^3+10*x)*exp(20)+( -2*x-5)*exp(4)^3+(6*x^2+15*x-10)*exp(4)^2+(-6*x^3-15*x^2+10*x-20)*exp(4)+2 *x^4+5*x^3+20*x-50)/(exp(x^2)^3+(-3*exp(4)+3*x)*exp(x^2)^2+(3*exp(4)^2-6*x *exp(4)+3*x^2)*exp(x^2)-exp(4)^3+3*x*exp(4)^2-3*x^2*exp(4)+x^3),x, algorit hm="giac")
Output:
(x^4 + 2*x^3*e^20 - 2*x^3*e^4 + 2*x^3*e^(x^2) + 5*x^3 - 4*x^2*e^24 + x^2*e ^8 - 10*x^2*e^4 + x^2*e^(2*x^2) + 4*x^2*e^(x^2 + 20) - 2*x^2*e^(x^2 + 4) + 10*x^2*e^(x^2) - 10*x^2 + 2*x*e^28 - 10*x*e^20 + 5*x*e^8 + 10*x*e^4 + 5*x *e^(2*x^2) + 2*x*e^(2*x^2 + 20) - 4*x*e^(x^2 + 24) - 10*x*e^(x^2 + 4) - 10 *x*e^(x^2) - 20*x + 10*e^24 + 20*e^4 - 10*e^(x^2 + 20) - 20*e^(x^2) + 25)/ (x^2 - 2*x*e^4 + 2*x*e^(x^2) + e^8 + e^(2*x^2) - 2*e^(x^2 + 4))
Timed out. \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\int \frac {20\,x+{\mathrm {e}}^{2\,x^2}\,\left (55\,x+{\mathrm {e}}^{20}\,\left (26\,x-6\,{\mathrm {e}}^4\right )+26\,x^2-{\mathrm {e}}^4\,\left (6\,x+15\right )-10\right )+{\mathrm {e}}^8\,\left (6\,x^2+15\,x-10\right )-{\mathrm {e}}^4\,\left (6\,x^3+15\,x^2-10\,x+20\right )+{\mathrm {e}}^{20}\,\left (10\,x-2\,{\mathrm {e}}^{12}+6\,x\,{\mathrm {e}}^8-{\mathrm {e}}^4\,\left (6\,x^2+10\right )+2\,x^3\right )+{\mathrm {e}}^{x^2}\,\left ({\mathrm {e}}^{20}\,\left (26\,x^2-32\,{\mathrm {e}}^4\,x+6\,{\mathrm {e}}^8+10\right )-{\mathrm {e}}^4\,\left (32\,x^2+70\,x-20\right )-110\,x+55\,x^2+26\,x^3+{\mathrm {e}}^8\,\left (6\,x+15\right )+20\right )+5\,x^3+2\,x^4+{\mathrm {e}}^{3\,x^2}\,\left (2\,x+2\,{\mathrm {e}}^{20}+5\right )-{\mathrm {e}}^{12}\,\left (2\,x+5\right )-50}{{\mathrm {e}}^{3\,x^2}-{\mathrm {e}}^{12}+{\mathrm {e}}^{x^2}\,\left (3\,x^2-6\,{\mathrm {e}}^4\,x+3\,{\mathrm {e}}^8\right )+3\,x\,{\mathrm {e}}^8-3\,x^2\,{\mathrm {e}}^4+x^3+{\mathrm {e}}^{2\,x^2}\,\left (3\,x-3\,{\mathrm {e}}^4\right )} \,d x \] Input:
int((20*x + exp(2*x^2)*(55*x + exp(20)*(26*x - 6*exp(4)) + 26*x^2 - exp(4) *(6*x + 15) - 10) + exp(8)*(15*x + 6*x^2 - 10) - exp(4)*(15*x^2 - 10*x + 6 *x^3 + 20) + exp(20)*(10*x - 2*exp(12) + 6*x*exp(8) - exp(4)*(6*x^2 + 10) + 2*x^3) + exp(x^2)*(exp(20)*(6*exp(8) - 32*x*exp(4) + 26*x^2 + 10) - exp( 4)*(70*x + 32*x^2 - 20) - 110*x + 55*x^2 + 26*x^3 + exp(8)*(6*x + 15) + 20 ) + 5*x^3 + 2*x^4 + exp(3*x^2)*(2*x + 2*exp(20) + 5) - exp(12)*(2*x + 5) - 50)/(exp(3*x^2) - exp(12) + exp(x^2)*(3*exp(8) - 6*x*exp(4) + 3*x^2) + 3* x*exp(8) - 3*x^2*exp(4) + x^3 + exp(2*x^2)*(3*x - 3*exp(4))),x)
Output:
int((20*x + exp(2*x^2)*(55*x + exp(20)*(26*x - 6*exp(4)) + 26*x^2 - exp(4) *(6*x + 15) - 10) + exp(8)*(15*x + 6*x^2 - 10) - exp(4)*(15*x^2 - 10*x + 6 *x^3 + 20) + exp(20)*(10*x - 2*exp(12) + 6*x*exp(8) - exp(4)*(6*x^2 + 10) + 2*x^3) + exp(x^2)*(exp(20)*(6*exp(8) - 32*x*exp(4) + 26*x^2 + 10) - exp( 4)*(70*x + 32*x^2 - 20) - 110*x + 55*x^2 + 26*x^3 + exp(8)*(6*x + 15) + 20 ) + 5*x^3 + 2*x^4 + exp(3*x^2)*(2*x + 2*exp(20) + 5) - exp(12)*(2*x + 5) - 50)/(exp(3*x^2) - exp(12) + exp(x^2)*(3*exp(8) - 6*x*exp(4) + 3*x^2) + 3* x*exp(8) - 3*x^2*exp(4) + x^3 + exp(2*x^2)*(3*x - 3*exp(4))), x)
Time = 31.08 (sec) , antiderivative size = 354, normalized size of antiderivative = 13.11 \[ \int \frac {-50+e^{12} (-5-2 x)+20 x+5 x^3+2 x^4+e^{3 x^2} \left (5+2 e^{20}+2 x\right )+e^8 \left (-10+15 x+6 x^2\right )+e^4 \left (-20+10 x-15 x^2-6 x^3\right )+e^{2 x^2} \left (-10+e^4 (-15-6 x)+55 x+26 x^2+e^{20} \left (-6 e^4+26 x\right )\right )+e^{20} \left (-2 e^{12}+10 x+6 e^8 x+2 x^3+e^4 \left (-10-6 x^2\right )\right )+e^{x^2} \left (20-110 x+55 x^2+26 x^3+e^8 (15+6 x)+e^4 \left (20-70 x-32 x^2\right )+e^{20} \left (10+6 e^8-32 e^4 x+26 x^2\right )\right )}{-e^{12}+e^{3 x^2}+3 e^8 x-3 e^4 x^2+x^3+e^{2 x^2} \left (-3 e^4+3 x\right )+e^{x^2} \left (3 e^8-6 e^4 x+3 x^2\right )} \, dx=\frac {-20 e^{4} x^{2}-20 e^{x^{2}} x +50 e^{4}-10 x^{2}+10 e^{4} x^{3}-20 e^{4} x -5 e^{20} x^{2}+4 e^{2 x^{2}} e^{24} x +2 e^{2 x^{2}} e^{4} x^{2}+10 e^{2 x^{2}} e^{4} x -8 e^{x^{2}} e^{28} x +8 e^{x^{2}} e^{24} x^{2}-10 e^{x^{2}} e^{20} x -4 e^{x^{2}} e^{8} x^{2}-20 e^{x^{2}} e^{8} x +4 e^{x^{2}} e^{4} x^{3}+20 e^{x^{2}} e^{4} x^{2}-20 e^{x^{2}} e^{4} x +2 e^{4} x^{4}-5 e^{2 x^{2}} e^{20}-10 e^{x^{2}} e^{24}+4 e^{32} x -8 e^{28} x^{2}+4 e^{24} x^{3}-10 e^{24} x +2 e^{12} x^{2}-4 e^{8} x^{3}-20 e^{8} x^{2}+20 e^{8} x -10 e^{2 x^{2}}+10 e^{12} x +15 e^{28}-20 e^{x^{2}} e^{4}+30 e^{8}}{2 e^{4} \left (e^{2 x^{2}}-2 e^{x^{2}} e^{4}+2 e^{x^{2}} x +e^{8}-2 e^{4} x +x^{2}\right )} \] Input:
int(((2*exp(20)+5+2*x)*exp(x^2)^3+((-6*exp(4)+26*x)*exp(20)+(-6*x-15)*exp( 4)+26*x^2+55*x-10)*exp(x^2)^2+((6*exp(4)^2-32*x*exp(4)+26*x^2+10)*exp(20)+ (6*x+15)*exp(4)^2+(-32*x^2-70*x+20)*exp(4)+26*x^3+55*x^2-110*x+20)*exp(x^2 )+(-2*exp(4)^3+6*x*exp(4)^2+(-6*x^2-10)*exp(4)+2*x^3+10*x)*exp(20)+(-2*x-5 )*exp(4)^3+(6*x^2+15*x-10)*exp(4)^2+(-6*x^3-15*x^2+10*x-20)*exp(4)+2*x^4+5 *x^3+20*x-50)/(exp(x^2)^3+(-3*exp(4)+3*x)*exp(x^2)^2+(3*exp(4)^2-6*x*exp(4 )+3*x^2)*exp(x^2)-exp(4)^3+3*x*exp(4)^2-3*x^2*exp(4)+x^3),x)
Output:
(4*e**(2*x**2)*e**24*x - 5*e**(2*x**2)*e**20 + 2*e**(2*x**2)*e**4*x**2 + 1 0*e**(2*x**2)*e**4*x - 10*e**(2*x**2) - 8*e**(x**2)*e**28*x + 8*e**(x**2)* e**24*x**2 - 10*e**(x**2)*e**24 - 10*e**(x**2)*e**20*x - 4*e**(x**2)*e**8* x**2 - 20*e**(x**2)*e**8*x + 4*e**(x**2)*e**4*x**3 + 20*e**(x**2)*e**4*x** 2 - 20*e**(x**2)*e**4*x - 20*e**(x**2)*e**4 - 20*e**(x**2)*x + 4*e**32*x - 8*e**28*x**2 + 15*e**28 + 4*e**24*x**3 - 10*e**24*x - 5*e**20*x**2 + 2*e* *12*x**2 + 10*e**12*x - 4*e**8*x**3 - 20*e**8*x**2 + 20*e**8*x + 30*e**8 + 2*e**4*x**4 + 10*e**4*x**3 - 20*e**4*x**2 - 20*e**4*x + 50*e**4 - 10*x**2 )/(2*e**4*(e**(2*x**2) - 2*e**(x**2)*e**4 + 2*e**(x**2)*x + e**8 - 2*e**4* x + x**2))