Integrand size = 244, antiderivative size = 28 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=40 x+\frac {3}{\left (\frac {1}{3} \left (3+e^3-x\right )+e^{x^2} x\right )^2} \] Output:
3/(1/3*exp(3)+1-1/3*x+exp(x^2)*x)^2+40*x
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(28)=56\).
Time = 3.73 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\frac {27+40 \left (3+e^3\right )^2 x+80 \left (3+e^3\right ) \left (-1+3 e^{x^2}\right ) x^2+40 \left (1-3 e^{x^2}\right )^2 x^3}{\left (3+e^3-x+3 e^{x^2} x\right )^2} \] Input:
Integrate[(1134 + 40*E^9 + E^6*(360 - 120*x) - 1080*x + 360*x^2 - 40*x^3 + 1080*E^(3*x^2)*x^3 + E^3*(1080 - 720*x + 120*x^2) + E^(2*x^2)*(3240*x^2 + 1080*E^3*x^2 - 1080*x^3) + E^x^2*(-162 + 3240*x + 360*E^6*x - 2484*x^2 + 360*x^3 + E^3*(2160*x - 720*x^2)))/(27 + E^9 + E^6*(9 - 3*x) - 27*x + 9*x^ 2 - x^3 + 27*E^(3*x^2)*x^3 + E^3*(27 - 18*x + 3*x^2) + E^(2*x^2)*(81*x^2 + 27*E^3*x^2 - 27*x^3) + E^x^2*(81*x + 9*E^6*x - 54*x^2 + 9*x^3 + E^3*(54*x - 18*x^2))),x]
Output:
(27 + 40*(3 + E^3)^2*x + 80*(3 + E^3)*(-1 + 3*E^x^2)*x^2 + 40*(1 - 3*E^x^2 )^2*x^3)/(3 + E^3 - x + 3*E^x^2*x)^2
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.
| \(\displaystyle \int \frac {-40 x^3+360 x^2+e^3 \left (120 x^2-720 x+1080\right )+1080 e^{3 x^2} x^3+e^{2 x^2} \left (-1080 x^3+1080 e^3 x^2+3240 x^2\right )+e^{x^2} \left (360 x^3-2484 x^2+e^3 \left (2160 x-720 x^2\right )+360 e^6 x+3240 x-162\right )-1080 x+e^6 (360-120 x)+40 e^9+1134}{-x^3+9 x^2+e^3 \left (3 x^2-18 x+27\right )+27 e^{3 x^2} x^3+e^{2 x^2} \left (-27 x^3+27 e^3 x^2+81 x^2\right )+e^{x^2} \left (9 x^3-54 x^2+e^3 \left (54 x-18 x^2\right )+9 e^6 x+81 x\right )-27 x+e^6 (9-3 x)+e^9+27} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-40 x^3+360 x^2+e^3 \left (120 x^2-720 x+1080\right )+1080 e^{3 x^2} x^3+e^{2 x^2} \left (-1080 x^3+1080 e^3 x^2+3240 x^2\right )+e^{x^2} \left (360 x^3-2484 x^2+e^3 \left (2160 x-720 x^2\right )+360 e^6 x+3240 x-162\right )-1080 x+e^6 (360-120 x)+1134 \left (1+\frac {20 e^9}{567}\right )}{\left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {54 \left (-2 x^2-1\right )}{x \left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^2}+\frac {54 \left (-2 x^3+2 \left (3+e^3\right ) x^2+e^3+3\right )}{x \left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^3}+40\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 108 \int \frac {x^2}{\left (-3 e^{x^2} x+x-3 \left (1+\frac {e^3}{3}\right )\right )^3}dx+54 \left (3+e^3\right ) \int \frac {1}{x \left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^3}dx+108 \left (3+e^3\right ) \int \frac {x}{\left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^3}dx-54 \int \frac {1}{x \left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^2}dx-108 \int \frac {x}{\left (3 e^{x^2} x-x+3 \left (1+\frac {e^3}{3}\right )\right )^2}dx+40 x\) |
Input:
Int[(1134 + 40*E^9 + E^6*(360 - 120*x) - 1080*x + 360*x^2 - 40*x^3 + 1080* E^(3*x^2)*x^3 + E^3*(1080 - 720*x + 120*x^2) + E^(2*x^2)*(3240*x^2 + 1080* E^3*x^2 - 1080*x^3) + E^x^2*(-162 + 3240*x + 360*E^6*x - 2484*x^2 + 360*x^ 3 + E^3*(2160*x - 720*x^2)))/(27 + E^9 + E^6*(9 - 3*x) - 27*x + 9*x^2 - x^ 3 + 27*E^(3*x^2)*x^3 + E^3*(27 - 18*x + 3*x^2) + E^(2*x^2)*(81*x^2 + 27*E^ 3*x^2 - 27*x^3) + E^x^2*(81*x + 9*E^6*x - 54*x^2 + 9*x^3 + E^3*(54*x - 18* x^2))),x]
Output:
$Aborted
Time = 3.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(40 x +\frac {27}{\left (3 \,{\mathrm e}^{x^{2}} x +{\mathrm e}^{3}-x +3\right )^{2}}\) | \(23\) |
| norman | \(\frac {\left (-120 \,{\mathrm e}^{6}-720 \,{\mathrm e}^{3}-1080\right ) x +\left (-240 \,{\mathrm e}^{3}-720\right ) x^{2} {\mathrm e}^{x^{2}}+\left (720 \,{\mathrm e}^{3}+2160\right ) x^{2} {\mathrm e}^{2 x^{2}}+\left (480 \,{\mathrm e}^{6}+2880 \,{\mathrm e}^{3}+4320\right ) x \,{\mathrm e}^{x^{2}}+40 x^{3}-240 x^{3} {\mathrm e}^{x^{2}}+360 x^{3} {\mathrm e}^{2 x^{2}}+2187+80 \,{\mathrm e}^{9}+720 \,{\mathrm e}^{6}+2160 \,{\mathrm e}^{3}}{\left (3 \,{\mathrm e}^{x^{2}} x +{\mathrm e}^{3}-x +3\right )^{2}}\) | \(123\) |
| parallelrisch | \(\frac {2187-1080 x +2160 x^{2} {\mathrm e}^{2 x^{2}}-240 x^{3} {\mathrm e}^{x^{2}}-720 x \,{\mathrm e}^{3}-120 x \,{\mathrm e}^{6}-720 x^{2} {\mathrm e}^{x^{2}}+4320 \,{\mathrm e}^{x^{2}} x +360 x^{3} {\mathrm e}^{2 x^{2}}+80 \,{\mathrm e}^{9}+40 x^{3}+2160 \,{\mathrm e}^{3}+720 \,{\mathrm e}^{6}+2880 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x +720 \,{\mathrm e}^{2 x^{2}} {\mathrm e}^{3} x^{2}+480 \,{\mathrm e}^{6} {\mathrm e}^{x^{2}} x -240 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x^{2}}{9 x^{2} {\mathrm e}^{2 x^{2}}+6 \,{\mathrm e}^{3} {\mathrm e}^{x^{2}} x -6 x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{6}-2 x \,{\mathrm e}^{3}+x^{2}+18 \,{\mathrm e}^{x^{2}} x +6 \,{\mathrm e}^{3}-6 x +9}\) | \(190\) |
Input:
int((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(3 60*x*exp(3)^2+(-720*x^2+2160*x)*exp(3)+360*x^3-2484*x^2+3240*x-162)*exp(x^ 2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40*x^3+36 0*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^ 2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3 )^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x,method=_ RETURNVERBOSE)
Output:
40*x+27/(3*exp(x^2)*x+exp(3)-x+3)^2
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.04 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} + 40 \, x^{3} - 240 \, x^{2} + 40 \, x e^{6} - 80 \, {\left (x^{2} - 3 \, x\right )} e^{3} - 240 \, {\left (x^{3} - x^{2} e^{3} - 3 \, x^{2}\right )} e^{\left (x^{2}\right )} + 360 \, x + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} + x^{2} - 2 \, {\left (x - 3\right )} e^{3} - 6 \, {\left (x^{2} - x e^{3} - 3 \, x\right )} e^{\left (x^{2}\right )} - 6 \, x + e^{6} + 9} \] Input:
integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2 )^2+(360*x*exp(3)^2+(-720*x^2+2160*x)*exp(3)+360*x^3-2484*x^2+3240*x-162)* exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40* x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)* exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2) +exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x, a lgorithm="fricas")
Output:
(360*x^3*e^(2*x^2) + 40*x^3 - 240*x^2 + 40*x*e^6 - 80*(x^2 - 3*x)*e^3 - 24 0*(x^3 - x^2*e^3 - 3*x^2)*e^(x^2) + 360*x + 27)/(9*x^2*e^(2*x^2) + x^2 - 2 *(x - 3)*e^3 - 6*(x^2 - x*e^3 - 3*x)*e^(x^2) - 6*x + e^6 + 9)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (22) = 44\).
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=40 x + \frac {27}{9 x^{2} e^{2 x^{2}} + x^{2} - 2 x e^{3} - 6 x + \left (- 6 x^{2} + 18 x + 6 x e^{3}\right ) e^{x^{2}} + 9 + 6 e^{3} + e^{6}} \] Input:
integrate((1080*x**3*exp(x**2)**3+(1080*x**2*exp(3)-1080*x**3+3240*x**2)*e xp(x**2)**2+(360*x*exp(3)**2+(-720*x**2+2160*x)*exp(3)+360*x**3-2484*x**2+ 3240*x-162)*exp(x**2)+40*exp(3)**3+(-120*x+360)*exp(3)**2+(120*x**2-720*x+ 1080)*exp(3)-40*x**3+360*x**2-1080*x+1134)/(27*x**3*exp(x**2)**3+(27*x**2* exp(3)-27*x**3+81*x**2)*exp(x**2)**2+(9*x*exp(3)**2+(-18*x**2+54*x)*exp(3) +9*x**3-54*x**2+81*x)*exp(x**2)+exp(3)**3+(-3*x+9)*exp(3)**2+(3*x**2-18*x+ 27)*exp(3)-x**3+9*x**2-27*x+27),x)
Output:
40*x + 27/(9*x**2*exp(2*x**2) + x**2 - 2*x*exp(3) - 6*x + (-6*x**2 + 18*x + 6*x*exp(3))*exp(x**2) + 9 + 6*exp(3) + exp(6))
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} + 40 \, x^{3} - 80 \, x^{2} {\left (e^{3} + 3\right )} + 40 \, x {\left (e^{6} + 6 \, e^{3} + 9\right )} - 240 \, {\left (x^{3} - x^{2} {\left (e^{3} + 3\right )}\right )} e^{\left (x^{2}\right )} + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} + x^{2} - 2 \, x {\left (e^{3} + 3\right )} - 6 \, {\left (x^{2} - x {\left (e^{3} + 3\right )}\right )} e^{\left (x^{2}\right )} + e^{6} + 6 \, e^{3} + 9} \] Input:
integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2 )^2+(360*x*exp(3)^2+(-720*x^2+2160*x)*exp(3)+360*x^3-2484*x^2+3240*x-162)* exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40* x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)* exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2) +exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x, a lgorithm="maxima")
Output:
(360*x^3*e^(2*x^2) + 40*x^3 - 80*x^2*(e^3 + 3) + 40*x*(e^6 + 6*e^3 + 9) - 240*(x^3 - x^2*(e^3 + 3))*e^(x^2) + 27)/(9*x^2*e^(2*x^2) + x^2 - 2*x*(e^3 + 3) - 6*(x^2 - x*(e^3 + 3))*e^(x^2) + e^6 + 6*e^3 + 9)
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (22) = 44\).
Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.64 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} - 240 \, x^{3} e^{\left (x^{2}\right )} + 40 \, x^{3} - 80 \, x^{2} e^{3} + 240 \, x^{2} e^{\left (x^{2} + 3\right )} + 720 \, x^{2} e^{\left (x^{2}\right )} - 240 \, x^{2} + 40 \, x e^{6} + 240 \, x e^{3} + 360 \, x + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} - 6 \, x^{2} e^{\left (x^{2}\right )} + x^{2} - 2 \, x e^{3} + 6 \, x e^{\left (x^{2} + 3\right )} + 18 \, x e^{\left (x^{2}\right )} - 6 \, x + e^{6} + 6 \, e^{3} + 9} \] Input:
integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2 )^2+(360*x*exp(3)^2+(-720*x^2+2160*x)*exp(3)+360*x^3-2484*x^2+3240*x-162)* exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40* x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)* exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2) +exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x, a lgorithm="giac")
Output:
(360*x^3*e^(2*x^2) - 240*x^3*e^(x^2) + 40*x^3 - 80*x^2*e^3 + 240*x^2*e^(x^ 2 + 3) + 720*x^2*e^(x^2) - 240*x^2 + 40*x*e^6 + 240*x*e^3 + 360*x + 27)/(9 *x^2*e^(2*x^2) - 6*x^2*e^(x^2) + x^2 - 2*x*e^3 + 6*x*e^(x^2 + 3) + 18*x*e^ (x^2) - 6*x + e^6 + 6*e^3 + 9)
Timed out. \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\int \frac {40\,{\mathrm {e}}^9-1080\,x+{\mathrm {e}}^3\,\left (120\,x^2-720\,x+1080\right )+{\mathrm {e}}^{2\,x^2}\,\left (1080\,x^2\,{\mathrm {e}}^3+3240\,x^2-1080\,x^3\right )+{\mathrm {e}}^{x^2}\,\left (3240\,x+{\mathrm {e}}^3\,\left (2160\,x-720\,x^2\right )+360\,x\,{\mathrm {e}}^6-2484\,x^2+360\,x^3-162\right )+1080\,x^3\,{\mathrm {e}}^{3\,x^2}+360\,x^2-40\,x^3-{\mathrm {e}}^6\,\left (120\,x-360\right )+1134}{{\mathrm {e}}^9-27\,x+{\mathrm {e}}^3\,\left (3\,x^2-18\,x+27\right )+{\mathrm {e}}^{2\,x^2}\,\left (27\,x^2\,{\mathrm {e}}^3+81\,x^2-27\,x^3\right )+{\mathrm {e}}^{x^2}\,\left (81\,x+{\mathrm {e}}^3\,\left (54\,x-18\,x^2\right )+9\,x\,{\mathrm {e}}^6-54\,x^2+9\,x^3\right )+27\,x^3\,{\mathrm {e}}^{3\,x^2}+9\,x^2-x^3-{\mathrm {e}}^6\,\left (3\,x-9\right )+27} \,d x \] Input:
int((40*exp(9) - 1080*x + exp(3)*(120*x^2 - 720*x + 1080) + exp(2*x^2)*(10 80*x^2*exp(3) + 3240*x^2 - 1080*x^3) + exp(x^2)*(3240*x + exp(3)*(2160*x - 720*x^2) + 360*x*exp(6) - 2484*x^2 + 360*x^3 - 162) + 1080*x^3*exp(3*x^2) + 360*x^2 - 40*x^3 - exp(6)*(120*x - 360) + 1134)/(exp(9) - 27*x + exp(3) *(3*x^2 - 18*x + 27) + exp(2*x^2)*(27*x^2*exp(3) + 81*x^2 - 27*x^3) + exp( x^2)*(81*x + exp(3)*(54*x - 18*x^2) + 9*x*exp(6) - 54*x^2 + 9*x^3) + 27*x^ 3*exp(3*x^2) + 9*x^2 - x^3 - exp(6)*(3*x - 9) + 27),x)
Output:
int((40*exp(9) - 1080*x + exp(3)*(120*x^2 - 720*x + 1080) + exp(2*x^2)*(10 80*x^2*exp(3) + 3240*x^2 - 1080*x^3) + exp(x^2)*(3240*x + exp(3)*(2160*x - 720*x^2) + 360*x*exp(6) - 2484*x^2 + 360*x^3 - 162) + 1080*x^3*exp(3*x^2) + 360*x^2 - 40*x^3 - exp(6)*(120*x - 360) + 1134)/(exp(9) - 27*x + exp(3) *(3*x^2 - 18*x + 27) + exp(2*x^2)*(27*x^2*exp(3) + 81*x^2 - 27*x^3) + exp( x^2)*(81*x + exp(3)*(54*x - 18*x^2) + 9*x*exp(6) - 54*x^2 + 9*x^3) + 27*x^ 3*exp(3*x^2) + 9*x^2 - x^3 - exp(6)*(3*x - 9) + 27), x)
Time = 0.20 (sec) , antiderivative size = 394, normalized size of antiderivative = 14.07 \[ \int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx=\frac {x \left (3402-2187 x +360 e^{2 x^{2}} e^{6} x^{2}+2160 e^{2 x^{2}} e^{3} x^{2}+240 e^{x^{2}} e^{9} x -240 e^{x^{2}} e^{6} x^{2}+2160 e^{x^{2}} e^{6} x -1440 e^{x^{2}} e^{3} x^{2}+6480 e^{x^{2}} e^{3} x -486 e^{x^{2}}+6642 e^{x^{2}} x -162 e^{x^{2}} e^{3}-80 e^{9} x +40 e^{6} x^{2}-720 e^{6} x +240 e^{3} x^{2}-2160 e^{x^{2}} x^{2}+40 e^{12}+360 x^{2}-2160 e^{3} x +3240 e^{2 x^{2}} x^{2}+2160 e^{6}-243 e^{2 x^{2}} x +480 e^{9}+4374 e^{3}\right )}{9 e^{2 x^{2}} e^{6} x^{2}+54 e^{2 x^{2}} e^{3} x^{2}+81 e^{2 x^{2}} x^{2}+6 e^{x^{2}} e^{9} x -6 e^{x^{2}} e^{6} x^{2}+54 e^{x^{2}} e^{6} x -36 e^{x^{2}} e^{3} x^{2}+162 e^{x^{2}} e^{3} x -54 e^{x^{2}} x^{2}+162 e^{x^{2}} x +e^{12}-2 e^{9} x +12 e^{9}+e^{6} x^{2}-18 e^{6} x +54 e^{6}+6 e^{3} x^{2}-54 e^{3} x +108 e^{3}+9 x^{2}-54 x +81} \] Input:
int((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(3 60*x*exp(3)^2+(-720*x^2+2160*x)*exp(3)+360*x^3-2484*x^2+3240*x-162)*exp(x^ 2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40*x^3+36 0*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^ 2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3 )^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x)
Output:
(x*(360*e**(2*x**2)*e**6*x**2 + 2160*e**(2*x**2)*e**3*x**2 + 3240*e**(2*x* *2)*x**2 - 243*e**(2*x**2)*x + 240*e**(x**2)*e**9*x - 240*e**(x**2)*e**6*x **2 + 2160*e**(x**2)*e**6*x - 1440*e**(x**2)*e**3*x**2 + 6480*e**(x**2)*e* *3*x - 162*e**(x**2)*e**3 - 2160*e**(x**2)*x**2 + 6642*e**(x**2)*x - 486*e **(x**2) + 40*e**12 - 80*e**9*x + 480*e**9 + 40*e**6*x**2 - 720*e**6*x + 2 160*e**6 + 240*e**3*x**2 - 2160*e**3*x + 4374*e**3 + 360*x**2 - 2187*x + 3 402))/(9*e**(2*x**2)*e**6*x**2 + 54*e**(2*x**2)*e**3*x**2 + 81*e**(2*x**2) *x**2 + 6*e**(x**2)*e**9*x - 6*e**(x**2)*e**6*x**2 + 54*e**(x**2)*e**6*x - 36*e**(x**2)*e**3*x**2 + 162*e**(x**2)*e**3*x - 54*e**(x**2)*x**2 + 162*e **(x**2)*x + e**12 - 2*e**9*x + 12*e**9 + e**6*x**2 - 18*e**6*x + 54*e**6 + 6*e**3*x**2 - 54*e**3*x + 108*e**3 + 9*x**2 - 54*x + 81)