Integrand size = 276, antiderivative size = 35 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\frac {4}{x+3 \left (e^{3 e^{2 x} x \left (e^{\frac {x}{2+2 x}}+x\right )^2}+x\right )} \] Output:
4/(4*x+3*exp(3*exp(x)^2*(x+exp(x/(2+2*x)))^2*x))
Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\frac {4}{3 e^{3 e^{2 x} x \left (e^{\frac {x}{2+2 x}}+x\right )^2}+4 x} \] Input:
Integrate[(-16 - 32*x - 16*x^2 + E^(2*x + E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(-108*x^2 - 288*x^3 - 252*x^4 - 72*x^5 + E^((2*x)/(2 + 2*x))*(-36 - 180*x - 180*x^2 - 72*x^3) + E^(x/(2 + 2*x))*( -144*x - 468*x^2 - 432*x^3 - 144*x^4)))/(16*x^2 + 32*x^3 + 16*x^4 + E^(2*E ^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(9 + 18* x + 9*x^2) + E^(E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(24*x + 48*x^2 + 24*x^3)),x]
Output:
4/(3*E^(3*E^(2*x)*x*(E^(x/(2 + 2*x)) + x)^2) + 4*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.
| \(\displaystyle \int \frac {\left (-72 x^5-252 x^4-288 x^3-108 x^2+e^{\frac {2 x}{2 x+2}} \left (-72 x^3-180 x^2-180 x-36\right )+e^{\frac {x}{2 x+2}} \left (-144 x^4-432 x^3-468 x^2-144 x\right )\right ) \exp \left (e^{2 x} \left (3 x^3+6 e^{\frac {x}{2 x+2}} x^2+3 e^{\frac {2 x}{2 x+2}} x\right )+2 x\right )-16 x^2-32 x-16}{\left (9 x^2+18 x+9\right ) \exp \left (2 e^{2 x} \left (3 x^3+6 e^{\frac {x}{2 x+2}} x^2+3 e^{\frac {2 x}{2 x+2}} x\right )\right )+\left (24 x^3+48 x^2+24 x\right ) \exp \left (e^{2 x} \left (3 x^3+6 e^{\frac {x}{2 x+2}} x^2+3 e^{\frac {2 x}{2 x+2}} x\right )\right )+16 x^4+32 x^3+16 x^2} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-16 x^2+36 e^{x \left (3 e^{2 x} \left (x+e^{\frac {x}{2 x+2}}\right )^2+2\right )} \left (-x^2 (2 x+3) (x+1)^2-e^{\frac {x}{x+1}} \left (2 x^3+5 x^2+5 x+1\right )-e^{\frac {x}{2 x+2}} x \left (4 x^3+12 x^2+13 x+4\right )\right )-32 x-16}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {16 x^2}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}+\frac {36 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2+2 x} \left (x+e^{\frac {x}{2 x+2}}\right ) \left (-2 x^4-2 e^{\frac {x}{2 x+2}} x^3-7 x^3-5 e^{\frac {x}{2 x+2}} x^2-8 x^2-5 e^{\frac {x}{2 x+2}} x-3 x-e^{\frac {x}{2 x+2}}\right )}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}-\frac {32 x}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}-\frac {16}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {16 x^2}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}+\frac {36 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2+2 x} \left (x+e^{\frac {x}{2 x+2}}\right ) \left (-2 x^4-2 e^{\frac {x}{2 x+2}} x^3-7 x^3-5 e^{\frac {x}{2 x+2}} x^2-8 x^2-5 e^{\frac {x}{2 x+2}} x-3 x-e^{\frac {x}{2 x+2}}\right )}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}-\frac {32 x}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}-\frac {16}{(x+1)^2 \left (4 x+3 e^{3 e^{2 x} x \left (x+e^{\frac {x}{2 x+2}}\right )^2}\right )^2}\right )dx\) |
Input:
Int[(-16 - 32*x - 16*x^2 + E^(2*x + E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E ^(x/(2 + 2*x))*x^2 + 3*x^3))*(-108*x^2 - 288*x^3 - 252*x^4 - 72*x^5 + E^(( 2*x)/(2 + 2*x))*(-36 - 180*x - 180*x^2 - 72*x^3) + E^(x/(2 + 2*x))*(-144*x - 468*x^2 - 432*x^3 - 144*x^4)))/(16*x^2 + 32*x^3 + 16*x^4 + E^(2*E^(2*x) *(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(9 + 18*x + 9* x^2) + E^(E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3 ))*(24*x + 48*x^2 + 24*x^3)),x]
Output:
$Aborted
Time = 30.80 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(\frac {4}{4 x +3 \,{\mathrm e}^{3 x \left (2 x \,{\mathrm e}^{\frac {x}{2+2 x}}+x^{2}+{\mathrm e}^{\frac {x}{1+x}}\right ) {\mathrm e}^{2 x}}}\) | \(43\) |
| parallelrisch | \(\frac {4}{4 x +3 \,{\mathrm e}^{\left (3 x \,{\mathrm e}^{\frac {x}{1+x}}+6 x^{2} {\mathrm e}^{\frac {x}{2+2 x}}+3 x^{3}\right ) {\mathrm e}^{2 x}}}\) | \(51\) |
Input:
int((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3-468*x^ 2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x* exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-16)/((9 *x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^ 2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3 *x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x,method=_RETURNVERBOSE)
Output:
4/(4*x+3*exp(3*x*(2*exp(1/2*x/(1+x))*x+x^2+exp(x/(1+x)))*exp(2*x)))
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\frac {4 \, e^{\left (2 \, x\right )}}{4 \, x e^{\left (2 \, x\right )} + 3 \, e^{\left (3 \, {\left (x^{3} + 2 \, x^{2} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )} + x e^{\left (\frac {x}{x + 1}\right )}\right )} e^{\left (2 \, x\right )} + 2 \, x\right )}} \] Input:
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3- 468*x^2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp ((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-1 6)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*e xp(x)^2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2 *x))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="fricas")
Output:
4*e^(2*x)/(4*x*e^(2*x) + 3*e^(3*(x^3 + 2*x^2*e^(1/2*x/(x + 1)) + x*e^(x/(x + 1)))*e^(2*x) + 2*x))
Time = 0.61 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\frac {4}{4 x + 3 e^{\left (3 x^{3} + 6 x^{2} e^{\frac {x}{2 x + 2}} + 3 x e^{\frac {2 x}{2 x + 2}}\right ) e^{2 x}}} \] Input:
integrate((((-72*x**3-180*x**2-180*x-36)*exp(x/(2+2*x))**2+(-144*x**4-432* x**3-468*x**2-144*x)*exp(x/(2+2*x))-72*x**5-252*x**4-288*x**3-108*x**2)*ex p(x)**2*exp((3*x*exp(x/(2+2*x))**2+6*x**2*exp(x/(2+2*x))+3*x**3)*exp(x)**2 )-16*x**2-32*x-16)/((9*x**2+18*x+9)*exp((3*x*exp(x/(2+2*x))**2+6*x**2*exp( x/(2+2*x))+3*x**3)*exp(x)**2)**2+(24*x**3+48*x**2+24*x)*exp((3*x*exp(x/(2+ 2*x))**2+6*x**2*exp(x/(2+2*x))+3*x**3)*exp(x)**2)+16*x**4+32*x**3+16*x**2) ,x)
Output:
4/(4*x + 3*exp((3*x**3 + 6*x**2*exp(x/(2*x + 2)) + 3*x*exp(2*x/(2*x + 2))) *exp(2*x)))
\[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\int { -\frac {4 \, {\left (4 \, x^{2} + 9 \, {\left (2 \, x^{5} + 7 \, x^{4} + 8 \, x^{3} + 3 \, x^{2} + {\left (2 \, x^{3} + 5 \, x^{2} + 5 \, x + 1\right )} e^{\left (\frac {x}{x + 1}\right )} + {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 4 \, x\right )} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )}\right )} e^{\left (3 \, {\left (x^{3} + 2 \, x^{2} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )} + x e^{\left (\frac {x}{x + 1}\right )}\right )} e^{\left (2 \, x\right )} + 2 \, x\right )} + 8 \, x + 4\right )}}{16 \, x^{4} + 32 \, x^{3} + 16 \, x^{2} + 9 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\left (6 \, {\left (x^{3} + 2 \, x^{2} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )} + x e^{\left (\frac {x}{x + 1}\right )}\right )} e^{\left (2 \, x\right )}\right )} + 24 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{\left (3 \, {\left (x^{3} + 2 \, x^{2} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )} + x e^{\left (\frac {x}{x + 1}\right )}\right )} e^{\left (2 \, x\right )}\right )}} \,d x } \] Input:
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3- 468*x^2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp ((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-1 6)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*e xp(x)^2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2 *x))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="maxima")
Output:
4*(3*(4*x^5*e^(1/2) + 12*x^4*e^(1/2) + 13*x^3*e^(1/2) + 4*x^2*e^(1/2))*e^( 2*x + 3/2/(x + 1)) + 3*(2*x^4*e + 5*x^3*e + 5*x^2*e + x*e)*e^(2*x + 1/(x + 1)) - (x^2 - 3*(2*x^6 + 7*x^5 + 8*x^4 + 3*x^3)*e^(2*x) + 2*x + 1)*e^(2/(x + 1)))/(3*(3*(4*x^5*e^(1/2) + 12*x^4*e^(1/2) + 13*x^3*e^(1/2) + 4*x^2*e^( 1/2))*e^(2*x + 3/2/(x + 1)) + (3*(2*x^4*e + 5*x^3*e + 5*x^2*e + x*e)*e^(2* x) - (x^2 - 3*(2*x^6 + 7*x^5 + 8*x^4 + 3*x^3)*e^(2*x) + 2*x + 1)*e^(1/(x + 1)))*e^(1/(x + 1)))*e^(3*x^3*e^(2*x) + 6*x^2*e^(2*x - 1/2/(x + 1) + 1/2) + 3*x*e^(2*x - 1/(x + 1) + 1)) + 12*(4*x^6*e^(1/2) + 12*x^5*e^(1/2) + 13*x ^4*e^(1/2) + 4*x^3*e^(1/2))*e^(2*x + 3/2/(x + 1)) + 4*(3*(2*x^5*e + 5*x^4* e + 5*x^3*e + x^2*e)*e^(2*x) - (x^3 + 2*x^2 - 3*(2*x^7 + 7*x^6 + 8*x^5 + 3 *x^4)*e^(2*x) + x)*e^(1/(x + 1)))*e^(1/(x + 1))) - 4*integrate(3*((3*(4*x^ 9*e^2 + 28*x^8*e^2 + 89*x^7*e^2 + 164*x^6*e^2 + 188*x^5*e^2 + 134*x^4*e^2 + 56*x^3*e^2 + 12*x^2*e^2 + x*e^2)*e^(4*x) + (3*(12*x^11*e + 88*x^10*e + 2 95*x^9*e + 581*x^8*e + 722*x^7*e + 572*x^6*e + 280*x^5*e + 77*x^4*e + 9*x^ 3*e)*e^(4*x) + (4*x^8*e + 28*x^7*e + 88*x^6*e + 156*x^5*e + 167*x^4*e + 10 8*x^3*e + 39*x^2*e + 6*x*e)*e^(2*x))*e^(1/(x + 1)))*e^(3/2/(x + 1)) - (3*( 4*x^9*e^2 + 28*x^8*e^2 + 89*x^7*e^2 + 164*x^6*e^2 + 188*x^5*e^2 + 134*x^4* e^2 + 56*x^3*e^2 + 12*x^2*e^2 + x*e^2)*e^(4*x + 1/(x + 1)) + (3*(12*x^11*e + 88*x^10*e + 295*x^9*e + 581*x^8*e + 722*x^7*e + 572*x^6*e + 280*x^5*e + 77*x^4*e + 9*x^3*e)*e^(4*x) + (4*x^8*e + 28*x^7*e + 88*x^6*e + 156*x^5...
Leaf count of result is larger than twice the leaf count of optimal. 27917 vs. \(2 (31) = 62\).
Time = 1.24 (sec) , antiderivative size = 27917, normalized size of antiderivative = 797.63 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3- 468*x^2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp ((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-1 6)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*e xp(x)^2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2 *x))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="giac")
Output:
4*(96*x^8*e^(3*x + 3/2*(2*x^2 + 3*x)/(x + 1)) + 72*x^7*e^(3*x + (3*x^4*e^( 2*x) + 3*x^3*e^(2*x) + 6*x^3*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 6*x^2*e^( x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 3*x^2*e^((2*x^2 + 3*x)/(x + 1)) + 2*x^2 + 3*x*e^((2*x^2 + 3*x)/(x + 1)) + 3*x)/(x + 1) + 1/2*(2*x^2 + 3*x)/(x + 1)) + 336*x^7*e^(3*x + 3/2*(2*x^2 + 3*x)/(x + 1)) + 72*x^7*e^(2*x + 1/2*(6*x^ 4*e^(2*x) + 6*x^3*e^(2*x) + 12*x^3*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 12* x^2*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 6*x^2*e^((2*x^2 + 3*x)/(x + 1)) + 4*x^2 + 6*x*e^((2*x^2 + 3*x)/(x + 1)) + 5*x)/(x + 1) + (2*x^2 + 3*x)/(x + 1)) + 192*x^7*e^(2*x + 2*(2*x^2 + 3*x)/(x + 1)) + 144*x^6*e^(3*x^3*e^(2*x) + 6*x^2*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 3*x*e^((2*x^2 + 3*x)/(x + 1)) + 2*x + 2*(2*x^2 + 3*x)/(x + 1)) + 252*x^6*e^(3*x + (3*x^4*e^(2*x) + 3*x^ 3*e^(2*x) + 6*x^3*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 6*x^2*e^(x + 1/2*(2* x^2 + 3*x)/(x + 1)) + 3*x^2*e^((2*x^2 + 3*x)/(x + 1)) + 2*x^2 + 3*x*e^((2* x^2 + 3*x)/(x + 1)) + 3*x)/(x + 1) + 1/2*(2*x^2 + 3*x)/(x + 1)) + 384*x^6* e^(3*x + 3/2*(2*x^2 + 3*x)/(x + 1)) + 54*x^6*e^(2*x + 1/2*(6*x^4*e^(2*x) + 6*x^3*e^(2*x) + 12*x^3*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 12*x^2*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 6*x^2*e^((2*x^2 + 3*x)/(x + 1)) + 4*x^2 + 6*x *e^((2*x^2 + 3*x)/(x + 1)) + 5*x)/(x + 1) + (3*x^4*e^(2*x) + 3*x^3*e^(2*x) + 6*x^3*e^(x + 1/2*(2*x^2 + 3*x)/(x + 1)) + 6*x^2*e^(x + 1/2*(2*x^2 + 3*x )/(x + 1)) + 3*x^2*e^((2*x^2 + 3*x)/(x + 1)) + 2*x^2 + 3*x*e^((2*x^2 + ...
Time = 4.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\frac {4}{4\,x+3\,{\mathrm {e}}^{6\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\frac {x}{2\,x+2}}}\,{\mathrm {e}}^{3\,x^3\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\frac {x}{x+1}}}} \] Input:
int(-(32*x + 16*x^2 + exp(2*x)*exp(exp(2*x)*(6*x^2*exp(x/(2*x + 2)) + 3*x* exp((2*x)/(2*x + 2)) + 3*x^3))*(exp(x/(2*x + 2))*(144*x + 468*x^2 + 432*x^ 3 + 144*x^4) + exp((2*x)/(2*x + 2))*(180*x + 180*x^2 + 72*x^3 + 36) + 108* x^2 + 288*x^3 + 252*x^4 + 72*x^5) + 16)/(exp(2*exp(2*x)*(6*x^2*exp(x/(2*x + 2)) + 3*x*exp((2*x)/(2*x + 2)) + 3*x^3))*(18*x + 9*x^2 + 9) + exp(exp(2* x)*(6*x^2*exp(x/(2*x + 2)) + 3*x*exp((2*x)/(2*x + 2)) + 3*x^3))*(24*x + 48 *x^2 + 24*x^3) + 16*x^2 + 32*x^3 + 16*x^4),x)
Output:
4/(4*x + 3*exp(6*x^2*exp(2*x)*exp(x/(2*x + 2)))*exp(3*x^3*exp(2*x))*exp(3* x*exp(2*x)*exp(x/(x + 1))))
\[ \int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} \left (-36-180 x-180 x^2-72 x^3\right )+e^{\frac {x}{2+2 x}} \left (-144 x-468 x^2-432 x^3-144 x^4\right )\right )}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (9+18 x+9 x^2\right )+e^{e^{2 x} \left (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3\right )} \left (24 x+48 x^2+24 x^3\right )} \, dx=\int \frac {\left (\left (-72 x^{3}-180 x^{2}-180 x -36\right ) \left ({\mathrm e}^{\frac {x}{2 x +2}}\right )^{2}+\left (-144 x^{4}-432 x^{3}-468 x^{2}-144 x \right ) {\mathrm e}^{\frac {x}{2 x +2}}-72 x^{5}-252 x^{4}-288 x^{3}-108 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{2} {\mathrm e}^{\left (3 x \left ({\mathrm e}^{\frac {x}{2 x +2}}\right )^{2}+6 x^{2} {\mathrm e}^{\frac {x}{2 x +2}}+3 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}}-16 x^{2}-32 x -16}{\left (9 x^{2}+18 x +9\right ) \left ({\mathrm e}^{\left (3 x \left ({\mathrm e}^{\frac {x}{2 x +2}}\right )^{2}+6 x^{2} {\mathrm e}^{\frac {x}{2 x +2}}+3 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}}\right )^{2}+\left (24 x^{3}+48 x^{2}+24 x \right ) {\mathrm e}^{\left (3 x \left ({\mathrm e}^{\frac {x}{2 x +2}}\right )^{2}+6 x^{2} {\mathrm e}^{\frac {x}{2 x +2}}+3 x^{3}\right ) \left ({\mathrm e}^{x}\right )^{2}}+16 x^{4}+32 x^{3}+16 x^{2}}d x \] Input:
int((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3-468*x^ 2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x* exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-16)/((9 *x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^ 2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3 *x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x)
Output:
int((((-72*x^3-180*x^2-180*x-36)*exp(x/(2+2*x))^2+(-144*x^4-432*x^3-468*x^ 2-144*x)*exp(x/(2+2*x))-72*x^5-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x* exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^2)-16*x^2-32*x-16)/((9 *x^2+18*x+9)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3*x^3)*exp(x)^ 2)^2+(24*x^3+48*x^2+24*x)*exp((3*x*exp(x/(2+2*x))^2+6*x^2*exp(x/(2+2*x))+3 *x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x)