Integrand size = 150, antiderivative size = 30 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{\left (-x+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} x+x^2\right )^2} \] Output:
2*exp((x^2+x*exp(1/4*exp(3+exp(x))+5/4)-x)^2)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{x^2 \left (-1+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}+x\right )^2} \] Input:
Integrate[E^(x^2 + E^((5 + E^(3 + E^x))/2)*x^2 - 2*x^3 + x^4 + E^((5 + E^( 3 + E^x))/4)*(-2*x^2 + 2*x^3))*(4*x - 12*x^2 + 8*x^3 + E^((5 + E^(3 + E^x) )/2)*(4*x + E^(3 + E^x + x)*x^2) + E^((5 + E^(3 + E^x))/4)*(-8*x + 12*x^2 + E^(3 + E^x + x)*(-x^2 + x^3))),x]
Output:
2*E^(x^2*(-1 + E^((5 + E^(3 + E^x))/4) + x)^2)
Time = 7.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {7292, 7257}
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.
| \(\displaystyle \int \left (8 x^3-12 x^2+e^{\frac {1}{2} \left (e^{e^x+3}+5\right )} \left (e^{x+e^x+3} x^2+4 x\right )+e^{\frac {1}{4} \left (e^{e^x+3}+5\right )} \left (12 x^2+e^{x+e^x+3} \left (x^3-x^2\right )-8 x\right )+4 x\right ) \exp \left (x^4-2 x^3+e^{\frac {1}{2} \left (e^{e^x+3}+5\right )} x^2+x^2+e^{\frac {1}{4} \left (e^{e^x+3}+5\right )} \left (2 x^3-2 x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int e^{x^2 \left (x+e^{\frac {e^{e^x+3}}{4}+\frac {5}{4}}-1\right )^2} \left (-x-e^{\frac {e^{e^x+3}}{4}+\frac {5}{4}}+1\right ) x \left (-e^{x+\frac {e^{e^x+3}}{4}+e^x+\frac {17}{4}} x-8 x-4 e^{\frac {e^{e^x+3}}{4}+\frac {5}{4}}+4\right )dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle 2 e^{\left (-x-e^{\frac {e^{e^x+3}}{4}+\frac {5}{4}}+1\right )^2 x^2}\) |
Input:
Int[E^(x^2 + E^((5 + E^(3 + E^x))/2)*x^2 - 2*x^3 + x^4 + E^((5 + E^(3 + E^ x))/4)*(-2*x^2 + 2*x^3))*(4*x - 12*x^2 + 8*x^3 + E^((5 + E^(3 + E^x))/2)*( 4*x + E^(3 + E^x + x)*x^2) + E^((5 + E^(3 + E^x))/4)*(-8*x + 12*x^2 + E^(3 + E^x + x)*(-x^2 + x^3))),x]
Output:
2*E^((1 - E^(5/4 + E^(3 + E^x)/4) - x)^2*x^2)
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).
Time = 0.92 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(2 \,{\mathrm e}^{x^{2} \left (2 x \,{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+x^{2}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{2}+\frac {5}{2}}-2 x +1\right )}\) | \(51\) |
| parallelrisch | \(2 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{2}+\frac {5}{2}}+\left (2 x^{3}-2 x^{2}\right ) {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+x^{4}-2 x^{3}+x^{2}}\) | \(54\) |
Input:
int(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x^3-x^2 )*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3-12*x^2 +4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp(3+exp (x))+5/4)+x^4-2*x^3+x^2),x,method=_RETURNVERBOSE)
Output:
2*exp(x^2*(2*x*exp(1/4*exp(3+exp(x))+5/4)+x^2-2*exp(1/4*exp(3+exp(x))+5/4) +exp(1/2*exp(3+exp(x))+5/2)-2*x+1))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 \, e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, {\left (e^{\left (x + e^{x} + 3\right )} + 5 \, e^{x}\right )} e^{\left (-x\right )}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, {\left (e^{\left (x + e^{x} + 3\right )} + 5 \, e^{x}\right )} e^{\left (-x\right )}\right )}\right )} \] Input:
integrate(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x ^3-x^2)*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3- 12*x^2+4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp (3+exp(x))+5/4)+x^4-2*x^3+x^2),x, algorithm="fricas")
Output:
2*e^(x^4 - 2*x^3 + x^2*e^(1/2*(e^(x + e^x + 3) + 5*e^x)*e^(-x)) + x^2 + 2* (x^3 - x^2)*e^(1/4*(e^(x + e^x + 3) + 5*e^x)*e^(-x)))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 3.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{x^{4} - 2 x^{3} + x^{2} e^{\frac {e^{e^{x} + 3}}{2} + \frac {5}{2}} + x^{2} + \left (2 x^{3} - 2 x^{2}\right ) e^{\frac {e^{e^{x} + 3}}{4} + \frac {5}{4}}} \] Input:
integrate(((x**2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)**2+( (x**3-x**2)*exp(x)*exp(3+exp(x))+12*x**2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8 *x**3-12*x**2+4*x)*exp(x**2*exp(1/4*exp(3+exp(x))+5/4)**2+(2*x**3-2*x**2)* exp(1/4*exp(3+exp(x))+5/4)+x**4-2*x**3+x**2),x)
Output:
2*exp(x**4 - 2*x**3 + x**2*exp(exp(exp(x) + 3)/2 + 5/2) + x**2 + (2*x**3 - 2*x**2)*exp(exp(exp(x) + 3)/4 + 5/4))
\[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=\int { {\left (8 \, x^{3} - 12 \, x^{2} + {\left (x^{2} e^{\left (x + e^{x} + 3\right )} + 4 \, x\right )} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + {\left (12 \, x^{2} + {\left (x^{3} - x^{2}\right )} e^{\left (x + e^{x} + 3\right )} - 8 \, x\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )} + 4 \, x\right )} e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )}\right )} \,d x } \] Input:
integrate(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x ^3-x^2)*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3- 12*x^2+4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp (3+exp(x))+5/4)+x^4-2*x^3+x^2),x, algorithm="maxima")
Output:
integrate((8*x^3 - 12*x^2 + (x^2*e^(x + e^x + 3) + 4*x)*e^(1/2*e^(e^x + 3) + 5/2) + (12*x^2 + (x^3 - x^2)*e^(x + e^x + 3) - 8*x)*e^(1/4*e^(e^x + 3) + 5/4) + 4*x)*e^(x^4 - 2*x^3 + x^2*e^(1/2*e^(e^x + 3) + 5/2) + x^2 + 2*(x^ 3 - x^2)*e^(1/4*e^(e^x + 3) + 5/4)), x)
\[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=\int { {\left (8 \, x^{3} - 12 \, x^{2} + {\left (x^{2} e^{\left (x + e^{x} + 3\right )} + 4 \, x\right )} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + {\left (12 \, x^{2} + {\left (x^{3} - x^{2}\right )} e^{\left (x + e^{x} + 3\right )} - 8 \, x\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )} + 4 \, x\right )} e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )}\right )} \,d x } \] Input:
integrate(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x ^3-x^2)*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3- 12*x^2+4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp (3+exp(x))+5/4)+x^4-2*x^3+x^2),x, algorithm="giac")
Output:
integrate((8*x^3 - 12*x^2 + (x^2*e^(x + e^x + 3) + 4*x)*e^(1/2*e^(e^x + 3) + 5/2) + (12*x^2 + (x^3 - x^2)*e^(x + e^x + 3) - 8*x)*e^(1/4*e^(e^x + 3) + 5/4) + 4*x)*e^(x^4 - 2*x^3 + x^2*e^(1/2*e^(e^x + 3) + 5/2) + x^2 + 2*(x^ 3 - x^2)*e^(1/4*e^(e^x + 3) + 5/4)), x)
Time = 3.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{5/2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{2}}}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{5/4}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{4}}}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{5/4}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{4}}}\,{\mathrm {e}}^{-2\,x^3} \] Input:
int(exp(x^2 - 2*x^3 + x^4 - exp(exp(exp(x) + 3)/4 + 5/4)*(2*x^2 - 2*x^3) + x^2*exp(exp(exp(x) + 3)/2 + 5/2))*(4*x + exp(exp(exp(x) + 3)/2 + 5/2)*(4* x + x^2*exp(exp(x) + 3)*exp(x)) - 12*x^2 + 8*x^3 - exp(exp(exp(x) + 3)/4 + 5/4)*(8*x - 12*x^2 + exp(exp(x) + 3)*exp(x)*(x^2 - x^3))),x)
Output:
2*exp(x^2)*exp(x^4)*exp(x^2*exp(5/2)*exp((exp(exp(x))*exp(3))/2))*exp(-2*x ^2*exp(5/4)*exp((exp(exp(x))*exp(3))/4))*exp(2*x^3*exp(5/4)*exp((exp(exp(x ))*exp(3))/4))*exp(-2*x^3)
\[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=\int \left (\left (x^{2} {\mathrm e}^{x} {\mathrm e}^{3+{\mathrm e}^{x}}+4 x \right ) \left ({\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}\right )^{2}+\left (\left (x^{3}-x^{2}\right ) {\mathrm e}^{x} {\mathrm e}^{3+{\mathrm e}^{x}}+12 x^{2}-8 x \right ) {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+8 x^{3}-12 x^{2}+4 x \right ) {\mathrm e}^{x^{2} \left ({\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}\right )^{2}+\left (2 x^{3}-2 x^{2}\right ) {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+x^{4}-2 x^{3}+x^{2}}d x \] Input:
int(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x^3-x^2 )*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3-12*x^2 +4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp(3+exp (x))+5/4)+x^4-2*x^3+x^2),x)
Output:
int(((x^2*exp(x)*exp(3+exp(x))+4*x)*exp(1/4*exp(3+exp(x))+5/4)^2+((x^3-x^2 )*exp(x)*exp(3+exp(x))+12*x^2-8*x)*exp(1/4*exp(3+exp(x))+5/4)+8*x^3-12*x^2 +4*x)*exp(x^2*exp(1/4*exp(3+exp(x))+5/4)^2+(2*x^3-2*x^2)*exp(1/4*exp(3+exp (x))+5/4)+x^4-2*x^3+x^2),x)