Integrand size = 415, antiderivative size = 37 \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=\left (-e^{x+e^{4+\frac {x}{x+\frac {3}{-e^x+x}}} x^2}+x^2\right )^2 \] Output:
(x^2-exp(x^2*exp(x/(x+3/(x-exp(x)))+4)+x))^2
Time = 1.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=\left (e^{x \left (1+e^{5-\frac {3}{3-e^x x+x^2}} x\right )}-x^2\right )^2 \] Input:
Integrate[(36*x^3 + 24*x^5 + 4*E^(2*x)*x^5 + 4*x^7 + E^x*(-24*x^4 - 8*x^6) + E^(2*x + 2*E^((-12 + 5*E^x*x - 5*x^2)/(-3 + E^x*x - x^2))*x^2)*(18 + 12 *x^2 + 2*E^(2*x)*x^2 + 2*x^4 + E^x*(-12*x - 4*x^3) + E^((-12 + 5*E^x*x - 5 *x^2)/(-3 + E^x*x - x^2))*(36*x + 36*x^3 + 4*E^(2*x)*x^3 + 4*x^5 + E^x*(-3 0*x^2 - 6*x^3 - 8*x^4))) + E^(x + E^((-12 + 5*E^x*x - 5*x^2)/(-3 + E^x*x - x^2))*x^2)*(-36*x - 18*x^2 - 24*x^3 - 12*x^4 - 4*x^5 - 2*x^6 + E^(2*x)*(- 4*x^3 - 2*x^4) + E^x*(24*x^2 + 12*x^3 + 8*x^4 + 4*x^5) + E^((-12 + 5*E^x*x - 5*x^2)/(-3 + E^x*x - x^2))*(-36*x^3 - 36*x^5 - 4*E^(2*x)*x^5 - 4*x^7 + E^x*(30*x^4 + 6*x^5 + 8*x^6))))/(9 + 6*x^2 + E^(2*x)*x^2 + x^4 + E^x*(-6*x - 2*x^3)),x]
Output:
(E^(x*(1 + E^(5 - 3/(3 - E^x*x + x^2))*x)) - x^2)^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.
| \(\displaystyle \int \frac {\left (2 x^4+e^x \left (-4 x^3-12 x\right )+2 e^{2 x} x^2+12 x^2+e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} \left (4 x^5+4 e^{2 x} x^3+36 x^3+e^x \left (-8 x^4-6 x^3-30 x^2\right )+36 x\right )+18\right ) \exp \left (2 e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} x^2+2 x\right )+\left (-2 x^6-4 x^5-12 x^4-24 x^3-18 x^2+e^{2 x} \left (-2 x^4-4 x^3\right )+e^x \left (4 x^5+8 x^4+12 x^3+24 x^2\right )+e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} \left (-4 x^7-4 e^{2 x} x^5-36 x^5-36 x^3+e^x \left (8 x^6+6 x^5+30 x^4\right )\right )-36 x\right ) \exp \left (e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} x^2+x\right )+4 x^7+4 e^{2 x} x^5+24 x^5+36 x^3+e^x \left (-8 x^6-24 x^4\right )}{x^4+e^x \left (-2 x^3-6 x\right )+e^{2 x} x^2+6 x^2+9} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x^4+e^x \left (-4 x^3-12 x\right )+2 e^{2 x} x^2+12 x^2+e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} \left (4 x^5+4 e^{2 x} x^3+36 x^3+e^x \left (-8 x^4-6 x^3-30 x^2\right )+36 x\right )+18\right ) \exp \left (2 e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} x^2+2 x\right )+\left (-2 x^6-4 x^5-12 x^4-24 x^3-18 x^2+e^{2 x} \left (-2 x^4-4 x^3\right )+e^x \left (4 x^5+8 x^4+12 x^3+24 x^2\right )+e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} \left (-4 x^7-4 e^{2 x} x^5-36 x^5-36 x^3+e^x \left (8 x^6+6 x^5+30 x^4\right )\right )-36 x\right ) \exp \left (e^{\frac {-5 x^2+5 e^x x-12}{-x^2+e^x x-3}} x^2+x\right )+4 x^7+4 e^{2 x} x^5+24 x^5+36 x^3+e^x \left (-8 x^6-24 x^4\right )}{\left (x^2-e^x x+3\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 x^7}{\left (x^2-e^x x+3\right )^2}-\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^6}{\left (-x^2+e^x x-3\right )^2}+\frac {4 e^{2 x} x^5}{\left (-x^2+e^x x-3\right )^2}-\frac {4 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^5}{\left (-x^2+e^x x-3\right )^2}+\frac {24 x^5}{\left (x^2-e^x x+3\right )^2}-\frac {8 e^x \left (x^2+3\right ) x^4}{\left (-x^2+e^x x-3\right )^2}-\frac {12 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^4}{\left (-x^2+e^x x-3\right )^2}+\frac {2 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )} x^4}{\left (x^2-e^x x+3\right )^2}-\frac {24 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^3}{\left (-x^2+e^x x-3\right )^2}+\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+3 x} (-x-2) x^3}{\left (x^2-e^x x+3\right )^2}+\frac {36 x^3}{\left (x^2-e^x x+3\right )^2}-\frac {18 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^2}{\left (-x^2+e^x x-3\right )^2}+\frac {12 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )} x^2}{\left (x^2-e^x x+3\right )^2}+\frac {2 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+2\right )} x^2}{\left (x^2-e^x x+3\right )^2}+\frac {4 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+2 x} (x+2) \left (x^2+3\right ) x^2}{\left (x^2-e^x x+3\right )^2}+\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+x+\frac {12}{x^2-e^x x+3}} \left (e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x}-x^2\right ) \left (2 x^4-4 e^x x^3-3 e^x x^2+2 e^{2 x} x^2+18 x^2-15 e^x x+18\right ) x}{\left (-x^2+e^x x-3\right )^2}-\frac {36 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x}{\left (-x^2+e^x x-3\right )^2}+\frac {4 e^{2 e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+3 x} \left (-x^2-3\right ) x}{\left (x^2-e^x x+3\right )^2}+\frac {18 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )}}{\left (-x^2+e^x x-3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {4 x^7}{\left (x^2-e^x x+3\right )^2}-\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^6}{\left (-x^2+e^x x-3\right )^2}+\frac {4 e^{2 x} x^5}{\left (-x^2+e^x x-3\right )^2}-\frac {4 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^5}{\left (-x^2+e^x x-3\right )^2}+\frac {24 x^5}{\left (x^2-e^x x+3\right )^2}-\frac {8 e^x \left (x^2+3\right ) x^4}{\left (-x^2+e^x x-3\right )^2}-\frac {12 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^4}{\left (-x^2+e^x x-3\right )^2}+\frac {2 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )} x^4}{\left (x^2-e^x x+3\right )^2}-\frac {24 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^3}{\left (-x^2+e^x x-3\right )^2}+\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+3 x} (-x-2) x^3}{\left (x^2-e^x x+3\right )^2}+\frac {36 x^3}{\left (x^2-e^x x+3\right )^2}-\frac {18 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x^2}{\left (-x^2+e^x x-3\right )^2}+\frac {12 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )} x^2}{\left (x^2-e^x x+3\right )^2}+\frac {2 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+2\right )} x^2}{\left (x^2-e^x x+3\right )^2}+\frac {4 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+2 x} (x+2) \left (x^2+3\right ) x^2}{\left (x^2-e^x x+3\right )^2}+\frac {2 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+x+\frac {12}{x^2-e^x x+3}} \left (e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x}-x^2\right ) \left (2 x^4-4 e^x x^3-3 e^x x^2+2 e^{2 x} x^2+18 x^2-15 e^x x+18\right ) x}{\left (-x^2+e^x x-3\right )^2}-\frac {36 e^{e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+x} x}{\left (-x^2+e^x x-3\right )^2}+\frac {4 e^{2 e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x^2+3 x} \left (-x^2-3\right ) x}{\left (x^2-e^x x+3\right )^2}+\frac {18 e^{2 x \left (e^{\frac {5 x^2}{x^2-e^x x+3}-\frac {5 e^x x}{x^2-e^x x+3}+\frac {12}{x^2-e^x x+3}} x+1\right )}}{\left (-x^2+e^x x-3\right )^2}\right )dx\) |
Input:
Int[(36*x^3 + 24*x^5 + 4*E^(2*x)*x^5 + 4*x^7 + E^x*(-24*x^4 - 8*x^6) + E^( 2*x + 2*E^((-12 + 5*E^x*x - 5*x^2)/(-3 + E^x*x - x^2))*x^2)*(18 + 12*x^2 + 2*E^(2*x)*x^2 + 2*x^4 + E^x*(-12*x - 4*x^3) + E^((-12 + 5*E^x*x - 5*x^2)/ (-3 + E^x*x - x^2))*(36*x + 36*x^3 + 4*E^(2*x)*x^3 + 4*x^5 + E^x*(-30*x^2 - 6*x^3 - 8*x^4))) + E^(x + E^((-12 + 5*E^x*x - 5*x^2)/(-3 + E^x*x - x^2)) *x^2)*(-36*x - 18*x^2 - 24*x^3 - 12*x^4 - 4*x^5 - 2*x^6 + E^(2*x)*(-4*x^3 - 2*x^4) + E^x*(24*x^2 + 12*x^3 + 8*x^4 + 4*x^5) + E^((-12 + 5*E^x*x - 5*x ^2)/(-3 + E^x*x - x^2))*(-36*x^3 - 36*x^5 - 4*E^(2*x)*x^5 - 4*x^7 + E^x*(3 0*x^4 + 6*x^5 + 8*x^6))))/(9 + 6*x^2 + E^(2*x)*x^2 + x^4 + E^x*(-6*x - 2*x ^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(34)=68\).
Time = 116.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.14
| method | result | size |
| risch | \(x^{4}-2 \,{\mathrm e}^{x \left (x \,{\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+1\right )} x^{2}+{\mathrm e}^{2 x \left (x \,{\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+1\right )}\) | \(79\) |
| parallelrisch | \(x^{4}-2 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+x} x^{2}+{\mathrm e}^{2 x^{2} {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+2 x}-27\) | \(81\) |
Input:
int((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x)*exp( (5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x)*exp(x )+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)^ 2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3)*exp(( 5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^5+8*x^ 4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x^2*exp ((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-24*x^4) *exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6*x^2+9 ),x,method=_RETURNVERBOSE)
Output:
x^4-2*exp(x*(x*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+1))*x^2+exp(2*x *(x*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+1))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (34) = 68\).
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.11 \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=x^{4} - 2 \, x^{2} e^{\left (x^{2} e^{\left (\frac {5 \, x^{2} - 5 \, x e^{x} + 12}{x^{2} - x e^{x} + 3}\right )} + x\right )} + e^{\left (2 \, x^{2} e^{\left (\frac {5 \, x^{2} - 5 \, x e^{x} + 12}{x^{2} - x e^{x} + 3}\right )} + 2 \, x\right )} \] Input:
integrate((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x )*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x) *exp(x)+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3 ))+x)^2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3) *exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^ 5+8*x^4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x ^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-2 4*x^4)*exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6 *x^2+9),x, algorithm="fricas")
Output:
x^4 - 2*x^2*e^(x^2*e^((5*x^2 - 5*x*e^x + 12)/(x^2 - x*e^x + 3)) + x) + e^( 2*x^2*e^((5*x^2 - 5*x*e^x + 12)/(x^2 - x*e^x + 3)) + 2*x)
Timed out. \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((((4*exp(x)**2*x**3+(-8*x**4-6*x**3-30*x**2)*exp(x)+4*x**5+36*x* *3+36*x)*exp((5*exp(x)*x-5*x**2-12)/(exp(x)*x-x**2-3))+2*exp(x)**2*x**2+(- 4*x**3-12*x)*exp(x)+2*x**4+12*x**2+18)*exp(x**2*exp((5*exp(x)*x-5*x**2-12) /(exp(x)*x-x**2-3))+x)**2+((-4*x**5*exp(x)**2+(8*x**6+6*x**5+30*x**4)*exp( x)-4*x**7-36*x**5-36*x**3)*exp((5*exp(x)*x-5*x**2-12)/(exp(x)*x-x**2-3))+( -2*x**4-4*x**3)*exp(x)**2+(4*x**5+8*x**4+12*x**3+24*x**2)*exp(x)-2*x**6-4* x**5-12*x**4-24*x**3-18*x**2-36*x)*exp(x**2*exp((5*exp(x)*x-5*x**2-12)/(ex p(x)*x-x**2-3))+x)+4*x**5*exp(x)**2+(-8*x**6-24*x**4)*exp(x)+4*x**7+24*x** 5+36*x**3)/(exp(x)**2*x**2+(-2*x**3-6*x)*exp(x)+x**4+6*x**2+9),x)
Output:
Timed out
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=x^{4} - 2 \, x^{2} e^{\left (x^{2} e^{\left (-\frac {3}{x^{2} - x e^{x} + 3} + 5\right )} + x\right )} + e^{\left (2 \, x^{2} e^{\left (-\frac {3}{x^{2} - x e^{x} + 3} + 5\right )} + 2 \, x\right )} \] Input:
integrate((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x )*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x) *exp(x)+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3 ))+x)^2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3) *exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^ 5+8*x^4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x ^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-2 4*x^4)*exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6 *x^2+9),x, algorithm="maxima")
Output:
x^4 - 2*x^2*e^(x^2*e^(-3/(x^2 - x*e^x + 3) + 5) + x) + e^(2*x^2*e^(-3/(x^2 - x*e^x + 3) + 5) + 2*x)
Exception generated. \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x )*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x) *exp(x)+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3 ))+x)^2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3) *exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^ 5+8*x^4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x ^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-2 4*x^4)*exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6 *x^2+9),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{16,[ 0,2,10,11]%%%}+%%%{-128,[0,2,9,12]%%%}+%%%{-48,[0,2,9,11]%%%}+%%%{-432,[0, 2,9,10]%%%}+
Time = 4.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.49 \[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx={\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{\frac {12}{x^2-x\,{\mathrm {e}}^x+3}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^x}{x^2-x\,{\mathrm {e}}^x+3}}\,{\mathrm {e}}^{\frac {5\,x^2}{x^2-x\,{\mathrm {e}}^x+3}}}+x^4-2\,x^2\,{\mathrm {e}}^x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {12}{x^2-x\,{\mathrm {e}}^x+3}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^x}{x^2-x\,{\mathrm {e}}^x+3}}\,{\mathrm {e}}^{\frac {5\,x^2}{x^2-x\,{\mathrm {e}}^x+3}}} \] Input:
int((4*x^5*exp(2*x) - exp(x)*(24*x^4 + 8*x^6) + exp(2*x + 2*x^2*exp((5*x^2 - 5*x*exp(x) + 12)/(x^2 - x*exp(x) + 3)))*(exp((5*x^2 - 5*x*exp(x) + 12)/ (x^2 - x*exp(x) + 3))*(36*x - exp(x)*(30*x^2 + 6*x^3 + 8*x^4) + 4*x^3*exp( 2*x) + 36*x^3 + 4*x^5) + 2*x^2*exp(2*x) - exp(x)*(12*x + 4*x^3) + 12*x^2 + 2*x^4 + 18) - exp(x + x^2*exp((5*x^2 - 5*x*exp(x) + 12)/(x^2 - x*exp(x) + 3)))*(36*x + exp(2*x)*(4*x^3 + 2*x^4) + exp((5*x^2 - 5*x*exp(x) + 12)/(x^ 2 - x*exp(x) + 3))*(4*x^5*exp(2*x) - exp(x)*(30*x^4 + 6*x^5 + 8*x^6) + 36* x^3 + 36*x^5 + 4*x^7) - exp(x)*(24*x^2 + 12*x^3 + 8*x^4 + 4*x^5) + 18*x^2 + 24*x^3 + 12*x^4 + 4*x^5 + 2*x^6) + 36*x^3 + 24*x^5 + 4*x^7)/(x^2*exp(2*x ) - exp(x)*(6*x + 2*x^3) + 6*x^2 + x^4 + 9),x)
Output:
exp(2*x)*exp(2*x^2*exp(12/(x^2 - x*exp(x) + 3))*exp(-(5*x*exp(x))/(x^2 - x *exp(x) + 3))*exp((5*x^2)/(x^2 - x*exp(x) + 3))) + x^4 - 2*x^2*exp(x)*exp( x^2*exp(12/(x^2 - x*exp(x) + 3))*exp(-(5*x*exp(x))/(x^2 - x*exp(x) + 3))*e xp((5*x^2)/(x^2 - x*exp(x) + 3)))
\[ \int \frac {36 x^3+24 x^5+4 e^{2 x} x^5+4 x^7+e^x \left (-24 x^4-8 x^6\right )+e^{2 x+2 e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (18+12 x^2+2 e^{2 x} x^2+2 x^4+e^x \left (-12 x-4 x^3\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (36 x+36 x^3+4 e^{2 x} x^3+4 x^5+e^x \left (-30 x^2-6 x^3-8 x^4\right )\right )\right )+e^{x+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} x^2} \left (-36 x-18 x^2-24 x^3-12 x^4-4 x^5-2 x^6+e^{2 x} \left (-4 x^3-2 x^4\right )+e^x \left (24 x^2+12 x^3+8 x^4+4 x^5\right )+e^{\frac {-12+5 e^x x-5 x^2}{-3+e^x x-x^2}} \left (-36 x^3-36 x^5-4 e^{2 x} x^5-4 x^7+e^x \left (30 x^4+6 x^5+8 x^6\right )\right )\right )}{9+6 x^2+e^{2 x} x^2+x^4+e^x \left (-6 x-2 x^3\right )} \, dx=\int \frac {\left (\left (4 \left ({\mathrm e}^{x}\right )^{2} x^{3}+\left (-8 x^{4}-6 x^{3}-30 x^{2}\right ) {\mathrm e}^{x}+4 x^{5}+36 x^{3}+36 x \right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+2 \left ({\mathrm e}^{x}\right )^{2} x^{2}+\left (-4 x^{3}-12 x \right ) {\mathrm e}^{x}+2 x^{4}+12 x^{2}+18\right ) \left ({\mathrm e}^{x^{2} {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+x}\right )^{2}+\left (\left (-4 x^{5} \left ({\mathrm e}^{x}\right )^{2}+\left (8 x^{6}+6 x^{5}+30 x^{4}\right ) {\mathrm e}^{x}-4 x^{7}-36 x^{5}-36 x^{3}\right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+\left (-2 x^{4}-4 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (4 x^{5}+8 x^{4}+12 x^{3}+24 x^{2}\right ) {\mathrm e}^{x}-2 x^{6}-4 x^{5}-12 x^{4}-24 x^{3}-18 x^{2}-36 x \right ) {\mathrm e}^{x^{2} {\mathrm e}^{\frac {5 \,{\mathrm e}^{x} x -5 x^{2}-12}{{\mathrm e}^{x} x -x^{2}-3}}+x}+4 x^{5} \left ({\mathrm e}^{x}\right )^{2}+\left (-8 x^{6}-24 x^{4}\right ) {\mathrm e}^{x}+4 x^{7}+24 x^{5}+36 x^{3}}{\left ({\mathrm e}^{x}\right )^{2} x^{2}+\left (-2 x^{3}-6 x \right ) {\mathrm e}^{x}+x^{4}+6 x^{2}+9}d x \] Input:
int((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x)*exp( (5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x)*exp(x )+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)^ 2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3)*exp(( 5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^5+8*x^ 4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x^2*exp ((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-24*x^4) *exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6*x^2+9 ),x)
Output:
int((((4*exp(x)^2*x^3+(-8*x^4-6*x^3-30*x^2)*exp(x)+4*x^5+36*x^3+36*x)*exp( (5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+2*exp(x)^2*x^2+(-4*x^3-12*x)*exp(x )+2*x^4+12*x^2+18)*exp(x^2*exp((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)^ 2+((-4*x^5*exp(x)^2+(8*x^6+6*x^5+30*x^4)*exp(x)-4*x^7-36*x^5-36*x^3)*exp(( 5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+(-2*x^4-4*x^3)*exp(x)^2+(4*x^5+8*x^ 4+12*x^3+24*x^2)*exp(x)-2*x^6-4*x^5-12*x^4-24*x^3-18*x^2-36*x)*exp(x^2*exp ((5*exp(x)*x-5*x^2-12)/(exp(x)*x-x^2-3))+x)+4*x^5*exp(x)^2+(-8*x^6-24*x^4) *exp(x)+4*x^7+24*x^5+36*x^3)/(exp(x)^2*x^2+(-2*x^3-6*x)*exp(x)+x^4+6*x^2+9 ),x)