Integrand size = 75, antiderivative size = 29 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=9 e^{2 x}-e^{-x-\left (-1+e^{-x}\right ) x^2} x \] Output:
9*exp(x)^2-x/exp(x^2*(1/exp(x)-1))/exp(x)
Time = 4.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=9 e^{2 x}-e^{-x \left (1+\left (-1+e^{-x}\right ) x\right )} x \] Input:
Integrate[E^(-2*x - (x^2 - E^x*x^2)/E^x)*(18*E^(4*x + (x^2 - E^x*x^2)/E^x) + 2*x^2 - x^3 + E^x*(-1 + x - 2*x^2)),x]
Output:
9*E^(2*x) - x/E^(x*(1 + (-1 + E^(-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.
\(\displaystyle \int e^{-e^{-x} \left (x^2-e^x x^2\right )-2 x} \left (-x^3+2 x^2+18 e^{e^{-x} \left (x^2-e^x x^2\right )+4 x}+e^x \left (-2 x^2+x-1\right )\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int e^{e^{-x} x \left (e^x x-x-2 e^x\right )} \left (-x^3+2 x^2+18 e^{e^{-x} \left (x^2-e^x x^2\right )+4 x}+e^x \left (-2 x^2+x-1\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (18 \exp \left (\left (e^{-x}-1\right ) x^2+e^{-x} \left (e^x x-x-2 e^x\right ) x+4 x\right )-e^{e^{-x} x \left (e^x x-x-2 e^x\right )} x^3+2 e^{e^{-x} x \left (e^x x-x-2 e^x\right )} x^2-e^{e^{-x} \left (e^x x-x-2 e^x\right ) x+x} \left (2 x^2-x+1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int e^{e^{-x} x \left (e^x x-x-2 e^x\right )} x^3dx+2 \int e^{e^{-x} x \left (e^x x-x-2 e^x\right )} x^2dx-2 \int e^{e^{-x} x \left (e^x x-x-e^x\right )} x^2dx-\int e^{e^{-x} x \left (e^x x-x-e^x\right )}dx+\int e^{e^{-x} x \left (e^x x-x-e^x\right )} xdx+\frac {18 \exp \left (-\left (\left (1-e^{-x}\right ) x^2\right )-e^{-x} \left (-e^x x+x+2 e^x\right ) x+4 x\right )}{-e^{-x} x^2-2 \left (1-e^{-x}\right ) x-e^{-x} \left (-e^x x+e^x+1\right ) x+e^{-x} \left (-e^x x+x+2 e^x\right ) x-e^{-x} \left (-e^x x+x+2 e^x\right )+4}\) |
Input:
Int[E^(-2*x - (x^2 - E^x*x^2)/E^x)*(18*E^(4*x + (x^2 - E^x*x^2)/E^x) + 2*x ^2 - x^3 + E^x*(-1 + x - 2*x^2)),x]
Output:
$Aborted
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
risch | \(9 \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{-x \left (x \,{\mathrm e}^{-x}-x +1\right )}\) | \(26\) |
parallelrisch | \(-{\mathrm e}^{-x} \left (-9 \,{\mathrm e}^{3 x} {\mathrm e}^{-x^{2} \left ({\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}+x \right ) {\mathrm e}^{x^{2} \left ({\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}\) | \(45\) |
Input:
int((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x ^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/exp(x)),x,method=_RETURNVERBOSE)
Output:
9*exp(2*x)-x*exp(-x*(x*exp(-x)-x+1))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=-x e^{\left (-{\left (x^{2} - {\left (x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (-x\right )} + x\right )} + 9 \, e^{\left (2 \, x\right )} \] Input:
integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x ^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/exp(x)),x, algorithm="fricas")
Output:
-x*e^(-(x^2 - (x^2 - 2*x)*e^x)*e^(-x) + x) + 9*e^(2*x)
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=- x e^{- x} e^{- \left (- x^{2} e^{x} + x^{2}\right ) e^{- x}} + 9 e^{2 x} \] Input:
integrate((18*exp(x)**4*exp((-exp(x)*x**2+x**2)/exp(x))+(-2*x**2+x-1)*exp( x)-x**3+2*x**2)/exp(x)**2/exp((-exp(x)*x**2+x**2)/exp(x)),x)
Output:
-x*exp(-x)*exp(-(-x**2*exp(x) + x**2)*exp(-x)) + 9*exp(2*x)
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=-x e^{\left (-x^{2} e^{\left (-x\right )} + x^{2} - x\right )} + 9 \, e^{\left (2 \, x\right )} \] Input:
integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x ^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/exp(x)),x, algorithm="maxima")
Output:
-x*e^(-x^2*e^(-x) + x^2 - x) + 9*e^(2*x)
\[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=\int { -{\left (x^{3} - 2 \, x^{2} + {\left (2 \, x^{2} - x + 1\right )} e^{x} - 18 \, e^{\left (-{\left (x^{2} e^{x} - x^{2}\right )} e^{\left (-x\right )} + 4 \, x\right )}\right )} e^{\left ({\left (x^{2} e^{x} - x^{2}\right )} e^{\left (-x\right )} - 2 \, x\right )} \,d x } \] Input:
integrate((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x ^3+2*x^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/exp(x)),x, algorithm="giac")
Output:
integrate(-(x^3 - 2*x^2 + (2*x^2 - x + 1)*e^x - 18*e^(-(x^2*e^x - x^2)*e^( -x) + 4*x))*e^((x^2*e^x - x^2)*e^(-x) - 2*x), x)
Time = 2.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=9\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-x}} \] Input:
int(exp(-2*x)*exp(exp(-x)*(x^2*exp(x) - x^2))*(18*exp(4*x)*exp(-exp(-x)*(x ^2*exp(x) - x^2)) - exp(x)*(2*x^2 - x + 1) + 2*x^2 - x^3),x)
Output:
9*exp(2*x) - x*exp(-x)*exp(x^2)*exp(-x^2*exp(-x))
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int e^{-2 x-e^{-x} \left (x^2-e^x x^2\right )} \left (18 e^{4 x+e^{-x} \left (x^2-e^x x^2\right )}+2 x^2-x^3+e^x \left (-1+x-2 x^2\right )\right ) \, dx=\frac {-e^{x^{2}} x +9 e^{\frac {3 e^{x} x +x^{2}}{e^{x}}}}{e^{\frac {e^{x} x +x^{2}}{e^{x}}}} \] Input:
int((18*exp(x)^4*exp((-exp(x)*x^2+x^2)/exp(x))+(-2*x^2+x-1)*exp(x)-x^3+2*x ^2)/exp(x)^2/exp((-exp(x)*x^2+x^2)/exp(x)),x)
Output:
( - e**(x**2)*x + 9*e**((3*e**x*x + x**2)/e**x))/e**((e**x*x + x**2)/e**x)