Integrand size = 79, antiderivative size = 29 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=3+e^{\frac {2 \left (1-\frac {e^{e^x}+x^2}{x}\right )}{3 (1+x)}} \] Output:
3+exp(2*(1-(x^2+exp(exp(x)))/x)/(3*x+3))
Time = 4.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{-\frac {2 \left (e^{e^x}+(-1+x) x\right )}{3 x (1+x)}} \] Input:
Integrate[(E^((-2*E^E^x + 2*x - 2*x^2)/(3*x + 3*x^2))*(-4*x^2 + E^E^x*(2 + 4*x + E^x*(-2*x - 2*x^2))))/(3*x^2 + 6*x^3 + 3*x^4),x]
Output:
E^((-2*(E^E^x + (-1 + x)*x))/(3*x*(1 + 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.
\(\displaystyle \int \frac {e^{\frac {-2 x^2+2 x-2 e^{e^x}}{3 x^2+3 x}} \left (e^{e^x} \left (e^x \left (-2 x^2-2 x\right )+4 x+2\right )-4 x^2\right )}{3 x^4+6 x^3+3 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{\frac {-2 x^2+2 x-2 e^{e^x}}{3 x^2+3 x}} \left (e^{e^x} \left (e^x \left (-2 x^2-2 x\right )+4 x+2\right )-4 x^2\right )}{x^2 \left (3 x^2+6 x+3\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{\frac {-2 x^2+2 x-2 e^{e^x}}{3 x^2+3 x}} \left (e^{e^x} \left (e^x \left (-2 x^2-2 x\right )+4 x+2\right )-4 x^2\right )}{x^2 \left (\sqrt {3} x+\sqrt {3}\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-\frac {2 \left (x^2-x+e^{e^x}\right )}{x (3 x+3)}} \left (e^{e^x} \left (e^x \left (-2 x^2-2 x\right )+4 x+2\right )-4 x^2\right )}{x^2 \left (\sqrt {3} x+\sqrt {3}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 e^{-\frac {2 \left (x^2-x+e^{e^x}\right )}{x (3 x+3)}} \left (2 x^2-2 e^{e^x} x-e^{e^x}\right )}{3 x^2 (x+1)^2}-\frac {2 e^{-\frac {2 \left (x^2-x+e^{e^x}\right )}{(3 x+3) x}+x+e^x}}{3 x (x+1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{3} \int \frac {e^{x+e^x-\frac {2 \left (x^2-x+e^{e^x}\right )}{(3 x+3) x}}}{-x-1}dx+\frac {2}{3} \int \frac {e^{e^x-\frac {2 \left (x^2-x+e^{e^x}\right )}{x (3 x+3)}}}{x^2}dx-\frac {2}{3} \int \frac {e^{x+e^x-\frac {2 \left (x^2-x+e^{e^x}\right )}{(3 x+3) x}}}{x}dx-\frac {4}{3} \int \frac {e^{-\frac {2 \left (x^2-x+e^{e^x}\right )}{x (3 x+3)}}}{(x+1)^2}dx-\frac {2}{3} \int \frac {e^{e^x-\frac {2 \left (x^2-x+e^{e^x}\right )}{x (3 x+3)}}}{(x+1)^2}dx\) |
Input:
Int[(E^((-2*E^E^x + 2*x - 2*x^2)/(3*x + 3*x^2))*(-4*x^2 + E^E^x*(2 + 4*x + E^x*(-2*x - 2*x^2))))/(3*x^2 + 6*x^3 + 3*x^4),x]
Output:
$Aborted
Time = 2.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76
method | result | size |
risch | \({\mathrm e}^{-\frac {2 \left (x^{2}+{\mathrm e}^{{\mathrm e}^{x}}-x \right )}{3 \left (1+x \right ) x}}\) | \(22\) |
parallelrisch | \({\mathrm e}^{-\frac {2 \left (x^{2}+{\mathrm e}^{{\mathrm e}^{x}}-x \right )}{3 \left (1+x \right ) x}}\) | \(22\) |
Input:
int((((-2*x^2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x^2)*exp((-2*exp(exp(x))-2* x^2+2*x)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x,method=_RETURNVERBOSE)
Output:
exp(-2/3*(x^2+exp(exp(x))-x)/(1+x)/x)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\left (-\frac {2 \, {\left (x^{2} - x + e^{\left (e^{x}\right )}\right )}}{3 \, {\left (x^{2} + x\right )}}\right )} \] Input:
integrate((((-2*x^2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x^2)*exp((-2*exp(exp( x))-2*x^2+2*x)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="fricas")
Output:
e^(-2/3*(x^2 - x + e^(e^x))/(x^2 + x))
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\frac {- 2 x^{2} + 2 x - 2 e^{e^{x}}}{3 x^{2} + 3 x}} \] Input:
integrate((((-2*x**2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x**2)*exp((-2*exp(ex p(x))-2*x**2+2*x)/(3*x**2+3*x))/(3*x**4+6*x**3+3*x**2),x)
Output:
exp((-2*x**2 + 2*x - 2*exp(exp(x)))/(3*x**2 + 3*x))
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\left (\frac {2 \, e^{\left (e^{x}\right )}}{3 \, {\left (x + 1\right )}} - \frac {2 \, e^{\left (e^{x}\right )}}{3 \, x} + \frac {4}{3 \, {\left (x + 1\right )}} - \frac {2}{3}\right )} \] Input:
integrate((((-2*x^2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x^2)*exp((-2*exp(exp( x))-2*x^2+2*x)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="maxima")
Output:
e^(2/3*e^(e^x)/(x + 1) - 2/3*e^(e^x)/x + 4/3/(x + 1) - 2/3)
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=e^{\left (-\frac {2 \, x^{2}}{3 \, {\left (x^{2} + x\right )}} + \frac {2 \, x}{3 \, {\left (x^{2} + x\right )}} - \frac {2 \, e^{\left (e^{x}\right )}}{3 \, {\left (x^{2} + x\right )}}\right )} \] Input:
integrate((((-2*x^2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x^2)*exp((-2*exp(exp( x))-2*x^2+2*x)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x, algorithm="giac")
Output:
e^(-2/3*x^2/(x^2 + x) + 2/3*x/(x^2 + x) - 2/3*e^(e^x)/(x^2 + x))
Time = 2.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx={\mathrm {e}}^{\frac {2}{3\,x+3}}\,{\mathrm {e}}^{-\frac {2\,x}{3\,x+3}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^x}}{3\,x^2+3\,x}} \] Input:
int((exp(-(2*exp(exp(x)) - 2*x + 2*x^2)/(3*x + 3*x^2))*(exp(exp(x))*(4*x - exp(x)*(2*x + 2*x^2) + 2) - 4*x^2))/(3*x^2 + 6*x^3 + 3*x^4),x)
Output:
exp(2/(3*x + 3))*exp(-(2*x)/(3*x + 3))*exp(-(2*exp(exp(x)))/(3*x + 3*x^2))
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {-2 e^{e^x}+2 x-2 x^2}{3 x+3 x^2}} \left (-4 x^2+e^{e^x} \left (2+4 x+e^x \left (-2 x-2 x^2\right )\right )\right )}{3 x^2+6 x^3+3 x^4} \, dx=\frac {e^{\frac {2}{3 x +3}}}{e^{\frac {2 e^{e^{x}}+2 x^{2}}{3 x^{2}+3 x}}} \] Input:
int((((-2*x^2-2*x)*exp(x)+4*x+2)*exp(exp(x))-4*x^2)*exp((-2*exp(exp(x))-2* x^2+2*x)/(3*x^2+3*x))/(3*x^4+6*x^3+3*x^2),x)
Output:
e**(2/(3*x + 3))/e**((2*e**(e**x) + 2*x**2)/(3*x**2 + 3*x))