Integrand size = 326, antiderivative size = 26 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=\left (x-x \left (e^2+x+\frac {x}{-6+x+e^{3 x} x}\right )\right )^2 \] Output:
(x-x*(exp(2)+x+x/(x+x*exp(3*x)-6)))^2
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(26)=52\).
Time = 10.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=x^2 \left (\left (-1+e^2\right )^2+2 \left (-1+e^2\right ) x+x^2+\frac {x^2}{\left (-6+x+e^{3 x} x\right )^2}+\frac {2 x \left (-1+e^2+x\right )}{-6+x+e^{3 x} x}\right ) \] Input:
Integrate[(-432*x + 1296*x^2 - 1224*x^3 + 456*x^4 - 72*x^5 + 4*x^6 + E^4*( -432*x + 216*x^2 - 36*x^3 + 2*x^4) + E^2*(864*x - 1512*x^2 + 660*x^3 - 108 *x^4 + 6*x^5) + E^(9*x)*(2*x^4 + 2*E^4*x^4 - 6*x^5 + 4*x^6 + E^2*(-4*x^4 + 6*x^5)) + E^(3*x)*(216*x^2 - 660*x^3 + 528*x^4 - 114*x^5 + 6*x^6 + E^4*(2 16*x^2 - 72*x^3 + 6*x^4) + E^2*(-432*x^2 + 732*x^3 - 184*x^4 + 12*x^5)) + E^(6*x)*(-36*x^3 + 110*x^4 - 78*x^5 + 6*x^6 + E^4*(-36*x^3 + 6*x^4) + E^2* (72*x^3 - 116*x^4 + 12*x^5)))/(-216 + 108*x - 18*x^2 + x^3 + E^(9*x)*x^3 + E^(3*x)*(108*x - 36*x^2 + 3*x^3) + E^(6*x)*(-18*x^2 + 3*x^3)),x]
Output:
x^2*((-1 + E^2)^2 + 2*(-1 + E^2)*x + x^2 + x^2/(-6 + x + E^(3*x)*x)^2 + (2 *x*(-1 + E^2 + x))/(-6 + x + E^(3*x)*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.
| \(\displaystyle \int \frac {4 x^6-72 x^5+456 x^4-1224 x^3+1296 x^2+e^{9 x} \left (4 x^6-6 x^5+2 e^4 x^4+2 x^4+e^2 \left (6 x^5-4 x^4\right )\right )+e^4 \left (2 x^4-36 x^3+216 x^2-432 x\right )+e^{6 x} \left (6 x^6-78 x^5+110 x^4-36 x^3+e^4 \left (6 x^4-36 x^3\right )+e^2 \left (12 x^5-116 x^4+72 x^3\right )\right )+e^2 \left (6 x^5-108 x^4+660 x^3-1512 x^2+864 x\right )+e^{3 x} \left (6 x^6-114 x^5+528 x^4-660 x^3+216 x^2+e^4 \left (6 x^4-72 x^3+216 x^2\right )+e^2 \left (12 x^5-184 x^4+732 x^3-432 x^2\right )\right )-432 x}{e^{9 x} x^3+x^3-18 x^2+e^{3 x} \left (3 x^3-36 x^2+108 x\right )+e^{6 x} \left (3 x^3-18 x^2\right )+108 x-216} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-4 x^6+72 x^5-456 x^4+1224 x^3-1296 x^2-e^{9 x} \left (4 x^6-6 x^5+2 e^4 x^4+2 x^4+e^2 \left (6 x^5-4 x^4\right )\right )-e^4 \left (2 x^4-36 x^3+216 x^2-432 x\right )-e^{6 x} \left (6 x^6-78 x^5+110 x^4-36 x^3+e^4 \left (6 x^4-36 x^3\right )+e^2 \left (12 x^5-116 x^4+72 x^3\right )\right )-e^2 \left (6 x^5-108 x^4+660 x^3-1512 x^2+864 x\right )-e^{3 x} \left (6 x^6-114 x^5+528 x^4-660 x^3+216 x^2+e^4 \left (6 x^4-72 x^3+216 x^2\right )+e^2 \left (12 x^5-184 x^4+732 x^3-432 x^2\right )\right )+432 x}{\left (-e^{3 x} x-x+6\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (3 x^2-3 \left (2-e^2\right ) x+2 \left (1-e^2\right )\right ) x^2}{-e^{3 x} x-x+6}+2 \left (2 x^2-3 \left (1-e^2\right ) x+\left (e^2-1\right )^2\right ) x+\frac {6 \left (x^2-6 x-2\right ) x^3}{\left (e^{3 x} x+x-6\right )^3}+\frac {2 \left (3 x^3-3 \left (8-e^2\right ) x^2+\left (13-18 e^2\right ) x+6 \left (1-e^2\right )\right ) x^2}{\left (-e^{3 x} x-x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int \frac {x^5}{\left (e^{3 x} x+x-6\right )^3}dx+6 \int \frac {x^5}{\left (e^{3 x} x+x-6\right )^2}dx-36 \int \frac {x^4}{\left (e^{3 x} x+x-6\right )^3}dx-6 \left (8-e^2\right ) \int \frac {x^4}{\left (e^{3 x} x+x-6\right )^2}dx-6 \int \frac {x^4}{e^{3 x} x+x-6}dx-12 \int \frac {x^3}{\left (e^{3 x} x+x-6\right )^3}dx+2 \left (13-18 e^2\right ) \int \frac {x^3}{\left (e^{3 x} x+x-6\right )^2}dx+6 \left (2-e^2\right ) \int \frac {x^3}{e^{3 x} x+x-6}dx+12 (1-e) (1+e) \int \frac {x^2}{\left (e^{3 x} x+x-6\right )^2}dx-4 (1-e) (1+e) \int \frac {x^2}{e^{3 x} x+x-6}dx+x^4-2 (1-e) (1+e) x^3+\left (1-e^2\right )^2 x^2\) |
Input:
Int[(-432*x + 1296*x^2 - 1224*x^3 + 456*x^4 - 72*x^5 + 4*x^6 + E^4*(-432*x + 216*x^2 - 36*x^3 + 2*x^4) + E^2*(864*x - 1512*x^2 + 660*x^3 - 108*x^4 + 6*x^5) + E^(9*x)*(2*x^4 + 2*E^4*x^4 - 6*x^5 + 4*x^6 + E^2*(-4*x^4 + 6*x^5 )) + E^(3*x)*(216*x^2 - 660*x^3 + 528*x^4 - 114*x^5 + 6*x^6 + E^4*(216*x^2 - 72*x^3 + 6*x^4) + E^2*(-432*x^2 + 732*x^3 - 184*x^4 + 12*x^5)) + E^(6*x )*(-36*x^3 + 110*x^4 - 78*x^5 + 6*x^6 + E^4*(-36*x^3 + 6*x^4) + E^2*(72*x^ 3 - 116*x^4 + 12*x^5)))/(-216 + 108*x - 18*x^2 + x^3 + E^(9*x)*x^3 + E^(3* x)*(108*x - 36*x^2 + 3*x^3) + E^(6*x)*(-18*x^2 + 3*x^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(24)=48\).
Time = 18.61 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54
| method | result | size |
| risch | \(x^{4}+2 x^{3} {\mathrm e}^{2}-2 x^{3}-2 x^{2} {\mathrm e}^{2}+x^{2} {\mathrm e}^{4}+x^{2}+\frac {\left (2 x \,{\mathrm e}^{2+3 x}+2 x^{2} {\mathrm e}^{3 x}+2 \,{\mathrm e}^{2} x +2 x^{2}-2 x \,{\mathrm e}^{3 x}-12 \,{\mathrm e}^{2}-13 x +12\right ) x^{3}}{\left (x +x \,{\mathrm e}^{3 x}-6\right )^{2}}\) | \(92\) |
| norman | \(\frac {x^{6}+\left (-12+2 \,{\mathrm e}^{2}\right ) x^{5}+\left (-72+84 \,{\mathrm e}^{2}-12 \,{\mathrm e}^{4}\right ) x^{3}+\left (48-24 \,{\mathrm e}^{2}+{\mathrm e}^{4}\right ) x^{4}+\left (36 \,{\mathrm e}^{4}-72 \,{\mathrm e}^{2}+36\right ) x^{2}+{\mathrm e}^{6 x} x^{6}+\left (-14+4 \,{\mathrm e}^{2}\right ) x^{5} {\mathrm e}^{3 x}+\left (-2+2 \,{\mathrm e}^{2}\right ) x^{5} {\mathrm e}^{6 x}+\left (-12-12 \,{\mathrm e}^{4}+24 \,{\mathrm e}^{2}\right ) x^{3} {\mathrm e}^{3 x}+\left (24-26 \,{\mathrm e}^{2}+2 \,{\mathrm e}^{4}\right ) x^{4} {\mathrm e}^{3 x}+\left ({\mathrm e}^{4}-2 \,{\mathrm e}^{2}+1\right ) x^{4} {\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} x^{6}}{\left (x +x \,{\mathrm e}^{3 x}-6\right )^{2}}\) | \(182\) |
| parallelrisch | \(\frac {-12 x^{3} {\mathrm e}^{3 x}+24 \,{\mathrm e}^{3 x} x^{4}-24 x^{4} {\mathrm e}^{2}-12 x^{3} {\mathrm e}^{4}+36 x^{2} {\mathrm e}^{4}-72 x^{2} {\mathrm e}^{2}+2 \,{\mathrm e}^{2} x^{5}+2 \,{\mathrm e}^{3 x} x^{6}+x^{4} {\mathrm e}^{4}+84 x^{3} {\mathrm e}^{2}-12 x^{5}+x^{6}+36 x^{2}-72 x^{3}+48 x^{4}+24 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x^{3}-26 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x^{4}+4 \,{\mathrm e}^{2} {\mathrm e}^{3 x} x^{5}+{\mathrm e}^{6 x} x^{6}-2 \,{\mathrm e}^{6 x} x^{5}+{\mathrm e}^{6 x} x^{4}-2 \,{\mathrm e}^{2} {\mathrm e}^{6 x} x^{4}-12 \,{\mathrm e}^{4} {\mathrm e}^{3 x} x^{3}+2 \,{\mathrm e}^{2} {\mathrm e}^{6 x} x^{5}+2 \,{\mathrm e}^{4} {\mathrm e}^{3 x} x^{4}+{\mathrm e}^{4} {\mathrm e}^{6 x} x^{4}-14 \,{\mathrm e}^{3 x} x^{5}}{x^{2} {\mathrm e}^{6 x}+2 x^{2} {\mathrm e}^{3 x}+x^{2}-12 x \,{\mathrm e}^{3 x}-12 x +36}\) | \(282\) |
Input:
int(((2*x^4*exp(2)^2+(6*x^5-4*x^4)*exp(2)+4*x^6-6*x^5+2*x^4)*exp(3*x)^3+(( 6*x^4-36*x^3)*exp(2)^2+(12*x^5-116*x^4+72*x^3)*exp(2)+6*x^6-78*x^5+110*x^4 -36*x^3)*exp(3*x)^2+((6*x^4-72*x^3+216*x^2)*exp(2)^2+(12*x^5-184*x^4+732*x ^3-432*x^2)*exp(2)+6*x^6-114*x^5+528*x^4-660*x^3+216*x^2)*exp(3*x)+(2*x^4- 36*x^3+216*x^2-432*x)*exp(2)^2+(6*x^5-108*x^4+660*x^3-1512*x^2+864*x)*exp( 2)+4*x^6-72*x^5+456*x^4-1224*x^3+1296*x^2-432*x)/(x^3*exp(3*x)^3+(3*x^3-18 *x^2)*exp(3*x)^2+(3*x^3-36*x^2+108*x)*exp(3*x)+x^3-18*x^2+108*x-216),x,met hod=_RETURNVERBOSE)
Output:
x^4+2*x^3*exp(2)-2*x^3-2*x^2*exp(2)+x^2*exp(4)+x^2+(2*x*exp(2+3*x)+2*x^2*e xp(3*x)+2*exp(2)*x+2*x^2-2*x*exp(3*x)-12*exp(2)-13*x+12)*x^3/(x+x*exp(3*x) -6)^2
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 7.23 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=\frac {x^{6} - 12 \, x^{5} + 48 \, x^{4} - 72 \, x^{3} + 36 \, x^{2} + {\left (x^{4} - 12 \, x^{3} + 36 \, x^{2}\right )} e^{4} + 2 \, {\left (x^{5} - 12 \, x^{4} + 42 \, x^{3} - 36 \, x^{2}\right )} e^{2} + {\left (x^{6} - 2 \, x^{5} + x^{4} e^{4} + x^{4} + 2 \, {\left (x^{5} - x^{4}\right )} e^{2}\right )} e^{\left (6 \, x\right )} + 2 \, {\left (x^{6} - 7 \, x^{5} + 12 \, x^{4} - 6 \, x^{3} + {\left (x^{4} - 6 \, x^{3}\right )} e^{4} + {\left (2 \, x^{5} - 13 \, x^{4} + 12 \, x^{3}\right )} e^{2}\right )} e^{\left (3 \, x\right )}}{x^{2} e^{\left (6 \, x\right )} + x^{2} + 2 \, {\left (x^{2} - 6 \, x\right )} e^{\left (3 \, x\right )} - 12 \, x + 36} \] Input:
integrate(((2*x^4*exp(2)^2+(6*x^5-4*x^4)*exp(2)+4*x^6-6*x^5+2*x^4)*exp(3*x )^3+((6*x^4-36*x^3)*exp(2)^2+(12*x^5-116*x^4+72*x^3)*exp(2)+6*x^6-78*x^5+1 10*x^4-36*x^3)*exp(3*x)^2+((6*x^4-72*x^3+216*x^2)*exp(2)^2+(12*x^5-184*x^4 +732*x^3-432*x^2)*exp(2)+6*x^6-114*x^5+528*x^4-660*x^3+216*x^2)*exp(3*x)+( 2*x^4-36*x^3+216*x^2-432*x)*exp(2)^2+(6*x^5-108*x^4+660*x^3-1512*x^2+864*x )*exp(2)+4*x^6-72*x^5+456*x^4-1224*x^3+1296*x^2-432*x)/(x^3*exp(3*x)^3+(3* x^3-18*x^2)*exp(3*x)^2+(3*x^3-36*x^2+108*x)*exp(3*x)+x^3-18*x^2+108*x-216) ,x, algorithm="fricas")
Output:
(x^6 - 12*x^5 + 48*x^4 - 72*x^3 + 36*x^2 + (x^4 - 12*x^3 + 36*x^2)*e^4 + 2 *(x^5 - 12*x^4 + 42*x^3 - 36*x^2)*e^2 + (x^6 - 2*x^5 + x^4*e^4 + x^4 + 2*( x^5 - x^4)*e^2)*e^(6*x) + 2*(x^6 - 7*x^5 + 12*x^4 - 6*x^3 + (x^4 - 6*x^3)* e^4 + (2*x^5 - 13*x^4 + 12*x^3)*e^2)*e^(3*x))/(x^2*e^(6*x) + x^2 + 2*(x^2 - 6*x)*e^(3*x) - 12*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.31 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=x^{4} + x^{3} \left (-2 + 2 e^{2}\right ) + x^{2} \left (- 2 e^{2} + 1 + e^{4}\right ) + \frac {2 x^{5} - 13 x^{4} + 2 x^{4} e^{2} - 12 x^{3} e^{2} + 12 x^{3} + \left (2 x^{5} - 2 x^{4} + 2 x^{4} e^{2}\right ) e^{3 x}}{x^{2} e^{6 x} + x^{2} - 12 x + \left (2 x^{2} - 12 x\right ) e^{3 x} + 36} \] Input:
integrate(((2*x**4*exp(2)**2+(6*x**5-4*x**4)*exp(2)+4*x**6-6*x**5+2*x**4)* exp(3*x)**3+((6*x**4-36*x**3)*exp(2)**2+(12*x**5-116*x**4+72*x**3)*exp(2)+ 6*x**6-78*x**5+110*x**4-36*x**3)*exp(3*x)**2+((6*x**4-72*x**3+216*x**2)*ex p(2)**2+(12*x**5-184*x**4+732*x**3-432*x**2)*exp(2)+6*x**6-114*x**5+528*x* *4-660*x**3+216*x**2)*exp(3*x)+(2*x**4-36*x**3+216*x**2-432*x)*exp(2)**2+( 6*x**5-108*x**4+660*x**3-1512*x**2+864*x)*exp(2)+4*x**6-72*x**5+456*x**4-1 224*x**3+1296*x**2-432*x)/(x**3*exp(3*x)**3+(3*x**3-18*x**2)*exp(3*x)**2+( 3*x**3-36*x**2+108*x)*exp(3*x)+x**3-18*x**2+108*x-216),x)
Output:
x**4 + x**3*(-2 + 2*exp(2)) + x**2*(-2*exp(2) + 1 + exp(4)) + (2*x**5 - 13 *x**4 + 2*x**4*exp(2) - 12*x**3*exp(2) + 12*x**3 + (2*x**5 - 2*x**4 + 2*x* *4*exp(2))*exp(3*x))/(x**2*exp(6*x) + x**2 - 12*x + (2*x**2 - 12*x)*exp(3* x) + 36)
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 6.08 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=\frac {x^{6} + 2 \, x^{5} {\left (e^{2} - 6\right )} + x^{4} {\left (e^{4} - 24 \, e^{2} + 48\right )} - 12 \, x^{3} {\left (e^{4} - 7 \, e^{2} + 6\right )} + 36 \, x^{2} {\left (e^{4} - 2 \, e^{2} + 1\right )} + {\left (x^{6} + 2 \, x^{5} {\left (e^{2} - 1\right )} + x^{4} {\left (e^{4} - 2 \, e^{2} + 1\right )}\right )} e^{\left (6 \, x\right )} + 2 \, {\left (x^{6} + x^{5} {\left (2 \, e^{2} - 7\right )} + x^{4} {\left (e^{4} - 13 \, e^{2} + 12\right )} - 6 \, x^{3} {\left (e^{4} - 2 \, e^{2} + 1\right )}\right )} e^{\left (3 \, x\right )}}{x^{2} e^{\left (6 \, x\right )} + x^{2} + 2 \, {\left (x^{2} - 6 \, x\right )} e^{\left (3 \, x\right )} - 12 \, x + 36} \] Input:
integrate(((2*x^4*exp(2)^2+(6*x^5-4*x^4)*exp(2)+4*x^6-6*x^5+2*x^4)*exp(3*x )^3+((6*x^4-36*x^3)*exp(2)^2+(12*x^5-116*x^4+72*x^3)*exp(2)+6*x^6-78*x^5+1 10*x^4-36*x^3)*exp(3*x)^2+((6*x^4-72*x^3+216*x^2)*exp(2)^2+(12*x^5-184*x^4 +732*x^3-432*x^2)*exp(2)+6*x^6-114*x^5+528*x^4-660*x^3+216*x^2)*exp(3*x)+( 2*x^4-36*x^3+216*x^2-432*x)*exp(2)^2+(6*x^5-108*x^4+660*x^3-1512*x^2+864*x )*exp(2)+4*x^6-72*x^5+456*x^4-1224*x^3+1296*x^2-432*x)/(x^3*exp(3*x)^3+(3* x^3-18*x^2)*exp(3*x)^2+(3*x^3-36*x^2+108*x)*exp(3*x)+x^3-18*x^2+108*x-216) ,x, algorithm="maxima")
Output:
(x^6 + 2*x^5*(e^2 - 6) + x^4*(e^4 - 24*e^2 + 48) - 12*x^3*(e^4 - 7*e^2 + 6 ) + 36*x^2*(e^4 - 2*e^2 + 1) + (x^6 + 2*x^5*(e^2 - 1) + x^4*(e^4 - 2*e^2 + 1))*e^(6*x) + 2*(x^6 + x^5*(2*e^2 - 7) + x^4*(e^4 - 13*e^2 + 12) - 6*x^3* (e^4 - 2*e^2 + 1))*e^(3*x))/(x^2*e^(6*x) + x^2 + 2*(x^2 - 6*x)*e^(3*x) - 1 2*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 255, normalized size of antiderivative = 9.81 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=\frac {x^{6} e^{\left (6 \, x\right )} + 2 \, x^{6} e^{\left (3 \, x\right )} + x^{6} + 2 \, x^{5} e^{2} - 2 \, x^{5} e^{\left (6 \, x\right )} - 14 \, x^{5} e^{\left (3 \, x\right )} + 2 \, x^{5} e^{\left (6 \, x + 2\right )} + 4 \, x^{5} e^{\left (3 \, x + 2\right )} - 12 \, x^{5} + x^{4} e^{4} - 24 \, x^{4} e^{2} + x^{4} e^{\left (6 \, x\right )} + 24 \, x^{4} e^{\left (3 \, x\right )} + x^{4} e^{\left (6 \, x + 4\right )} - 2 \, x^{4} e^{\left (6 \, x + 2\right )} + 2 \, x^{4} e^{\left (3 \, x + 4\right )} - 26 \, x^{4} e^{\left (3 \, x + 2\right )} + 48 \, x^{4} - 12 \, x^{3} e^{4} + 84 \, x^{3} e^{2} - 12 \, x^{3} e^{\left (3 \, x\right )} - 12 \, x^{3} e^{\left (3 \, x + 4\right )} + 24 \, x^{3} e^{\left (3 \, x + 2\right )} - 72 \, x^{3} + 36 \, x^{2} e^{4} - 72 \, x^{2} e^{2} + 36 \, x^{2}}{x^{2} e^{\left (6 \, x\right )} + 2 \, x^{2} e^{\left (3 \, x\right )} + x^{2} - 12 \, x e^{\left (3 \, x\right )} - 12 \, x + 36} \] Input:
integrate(((2*x^4*exp(2)^2+(6*x^5-4*x^4)*exp(2)+4*x^6-6*x^5+2*x^4)*exp(3*x )^3+((6*x^4-36*x^3)*exp(2)^2+(12*x^5-116*x^4+72*x^3)*exp(2)+6*x^6-78*x^5+1 10*x^4-36*x^3)*exp(3*x)^2+((6*x^4-72*x^3+216*x^2)*exp(2)^2+(12*x^5-184*x^4 +732*x^3-432*x^2)*exp(2)+6*x^6-114*x^5+528*x^4-660*x^3+216*x^2)*exp(3*x)+( 2*x^4-36*x^3+216*x^2-432*x)*exp(2)^2+(6*x^5-108*x^4+660*x^3-1512*x^2+864*x )*exp(2)+4*x^6-72*x^5+456*x^4-1224*x^3+1296*x^2-432*x)/(x^3*exp(3*x)^3+(3* x^3-18*x^2)*exp(3*x)^2+(3*x^3-36*x^2+108*x)*exp(3*x)+x^3-18*x^2+108*x-216) ,x, algorithm="giac")
Output:
(x^6*e^(6*x) + 2*x^6*e^(3*x) + x^6 + 2*x^5*e^2 - 2*x^5*e^(6*x) - 14*x^5*e^ (3*x) + 2*x^5*e^(6*x + 2) + 4*x^5*e^(3*x + 2) - 12*x^5 + x^4*e^4 - 24*x^4* e^2 + x^4*e^(6*x) + 24*x^4*e^(3*x) + x^4*e^(6*x + 4) - 2*x^4*e^(6*x + 2) + 2*x^4*e^(3*x + 4) - 26*x^4*e^(3*x + 2) + 48*x^4 - 12*x^3*e^4 + 84*x^3*e^2 - 12*x^3*e^(3*x) - 12*x^3*e^(3*x + 4) + 24*x^3*e^(3*x + 2) - 72*x^3 + 36* x^2*e^4 - 72*x^2*e^2 + 36*x^2)/(x^2*e^(6*x) + 2*x^2*e^(3*x) + x^2 - 12*x*e ^(3*x) - 12*x + 36)
Time = 3.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.62 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=x^3\,\left (2\,{\mathrm {e}}^2-2\right )+x^2\,{\left ({\mathrm {e}}^2-1\right )}^2+x^4-\frac {2\,\left (x^5\,{\mathrm {e}}^2-6\,x^4\,{\mathrm {e}}^2-2\,x^3\,{\mathrm {e}}^2+2\,x^3+4\,x^4-7\,x^5+x^6\right )}{\left (-x^2+6\,x+2\right )\,\left (x+x\,{\mathrm {e}}^{3\,x}-6\right )}+\frac {-x^7+6\,x^6+2\,x^5}{x\,\left (-x^2+6\,x+2\right )\,\left ({\left (x-6\right )}^2+x^2\,{\mathrm {e}}^{6\,x}+2\,x\,{\mathrm {e}}^{3\,x}\,\left (x-6\right )\right )} \] Input:
int((exp(9*x)*(2*x^4*exp(4) - exp(2)*(4*x^4 - 6*x^5) + 2*x^4 - 6*x^5 + 4*x ^6) - exp(6*x)*(exp(4)*(36*x^3 - 6*x^4) - exp(2)*(72*x^3 - 116*x^4 + 12*x^ 5) + 36*x^3 - 110*x^4 + 78*x^5 - 6*x^6) - exp(4)*(432*x - 216*x^2 + 36*x^3 - 2*x^4) - 432*x + exp(3*x)*(exp(4)*(216*x^2 - 72*x^3 + 6*x^4) + 216*x^2 - 660*x^3 + 528*x^4 - 114*x^5 + 6*x^6 - exp(2)*(432*x^2 - 732*x^3 + 184*x^ 4 - 12*x^5)) + exp(2)*(864*x - 1512*x^2 + 660*x^3 - 108*x^4 + 6*x^5) + 129 6*x^2 - 1224*x^3 + 456*x^4 - 72*x^5 + 4*x^6)/(108*x + exp(3*x)*(108*x - 36 *x^2 + 3*x^3) - exp(6*x)*(18*x^2 - 3*x^3) + x^3*exp(9*x) - 18*x^2 + x^3 - 216),x)
Output:
x^3*(2*exp(2) - 2) + x^2*(exp(2) - 1)^2 + x^4 - (2*(x^5*exp(2) - 6*x^4*exp (2) - 2*x^3*exp(2) + 2*x^3 + 4*x^4 - 7*x^5 + x^6))/((6*x - x^2 + 2)*(x + x *exp(3*x) - 6)) + (2*x^5 + 6*x^6 - x^7)/(x*(6*x - x^2 + 2)*((x - 6)^2 + x^ 2*exp(6*x) + 2*x*exp(3*x)*(x - 6)))
Time = 0.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 10.35 \[ \int \frac {-432 x+1296 x^2-1224 x^3+456 x^4-72 x^5+4 x^6+e^4 \left (-432 x+216 x^2-36 x^3+2 x^4\right )+e^2 \left (864 x-1512 x^2+660 x^3-108 x^4+6 x^5\right )+e^{9 x} \left (2 x^4+2 e^4 x^4-6 x^5+4 x^6+e^2 \left (-4 x^4+6 x^5\right )\right )+e^{3 x} \left (216 x^2-660 x^3+528 x^4-114 x^5+6 x^6+e^4 \left (216 x^2-72 x^3+6 x^4\right )+e^2 \left (-432 x^2+732 x^3-184 x^4+12 x^5\right )\right )+e^{6 x} \left (-36 x^3+110 x^4-78 x^5+6 x^6+e^4 \left (-36 x^3+6 x^4\right )+e^2 \left (72 x^3-116 x^4+12 x^5\right )\right )}{-216+108 x-18 x^2+x^3+e^{9 x} x^3+e^{3 x} \left (108 x-36 x^2+3 x^3\right )+e^{6 x} \left (-18 x^2+3 x^3\right )} \, dx=\frac {x^{2} \left (36-72 x -24 e^{2} x^{2}+84 e^{2} x +e^{4} x^{2}+36 e^{4}-72 e^{2}+e^{6 x} e^{4} x^{2}+48 x^{2}-12 e^{4} x -12 e^{3 x} x -12 x^{3}+e^{6 x} x^{2}+2 e^{6 x} e^{2} x^{3}-2 e^{6 x} e^{2} x^{2}+2 e^{3 x} e^{4} x^{2}-12 e^{3 x} e^{4} x +4 e^{3 x} e^{2} x^{3}-26 e^{3 x} e^{2} x^{2}+24 e^{3 x} e^{2} x +e^{6 x} x^{4}-2 e^{6 x} x^{3}+2 e^{3 x} x^{4}-14 e^{3 x} x^{3}+24 e^{3 x} x^{2}+x^{4}+2 e^{2} x^{3}\right )}{e^{6 x} x^{2}+2 e^{3 x} x^{2}-12 e^{3 x} x +x^{2}-12 x +36} \] Input:
int(((2*x^4*exp(2)^2+(6*x^5-4*x^4)*exp(2)+4*x^6-6*x^5+2*x^4)*exp(3*x)^3+(( 6*x^4-36*x^3)*exp(2)^2+(12*x^5-116*x^4+72*x^3)*exp(2)+6*x^6-78*x^5+110*x^4 -36*x^3)*exp(3*x)^2+((6*x^4-72*x^3+216*x^2)*exp(2)^2+(12*x^5-184*x^4+732*x ^3-432*x^2)*exp(2)+6*x^6-114*x^5+528*x^4-660*x^3+216*x^2)*exp(3*x)+(2*x^4- 36*x^3+216*x^2-432*x)*exp(2)^2+(6*x^5-108*x^4+660*x^3-1512*x^2+864*x)*exp( 2)+4*x^6-72*x^5+456*x^4-1224*x^3+1296*x^2-432*x)/(x^3*exp(3*x)^3+(3*x^3-18 *x^2)*exp(3*x)^2+(3*x^3-36*x^2+108*x)*exp(3*x)+x^3-18*x^2+108*x-216),x)
Output:
(x**2*(e**(6*x)*e**4*x**2 + 2*e**(6*x)*e**2*x**3 - 2*e**(6*x)*e**2*x**2 + e**(6*x)*x**4 - 2*e**(6*x)*x**3 + e**(6*x)*x**2 + 2*e**(3*x)*e**4*x**2 - 1 2*e**(3*x)*e**4*x + 4*e**(3*x)*e**2*x**3 - 26*e**(3*x)*e**2*x**2 + 24*e**( 3*x)*e**2*x + 2*e**(3*x)*x**4 - 14*e**(3*x)*x**3 + 24*e**(3*x)*x**2 - 12*e **(3*x)*x + e**4*x**2 - 12*e**4*x + 36*e**4 + 2*e**2*x**3 - 24*e**2*x**2 + 84*e**2*x - 72*e**2 + x**4 - 12*x**3 + 48*x**2 - 72*x + 36))/(e**(6*x)*x* *2 + 2*e**(3*x)*x**2 - 12*e**(3*x)*x + x**2 - 12*x + 36)