Integrand size = 72, antiderivative size = 31 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=-4+x^2 \left (e^{3 \left (-x+x^2 \left (-x+e^{-x} x\right )\right )}+x\right ) \] Output:
(x+exp(3*x^2*(x/exp(x)-x)-3*x))*x^2-4
Time = 5.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^2 \left (e^{-3 x+\left (-3+3 e^{-x}\right ) x^3}+x\right ) \] Input:
Integrate[(3*E^x*x^2 + E^((3*x^3 + E^x*(-3*x - 3*x^3))/E^x)*(9*x^4 - 3*x^5 + E^x*(2*x - 3*x^2 - 9*x^4)))/E^x,x]
Output:
x^2*(E^(-3*x + (-3 + 3/E^x)*x^3) + 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.
\(\displaystyle \int e^{-x} \left (\left (-3 x^5+9 x^4+e^x \left (-9 x^4-3 x^2+2 x\right )\right ) \exp \left (e^{-x} \left (3 x^3+e^x \left (-3 x^3-3 x\right )\right )\right )+3 e^x x^2\right ) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (3 x^2-e^{-3 \left (\left (1-e^{-x}\right ) x^2+1\right ) x-x} x \left (3 x^4+9 e^x x^3-9 x^3+3 e^x x-2 e^x\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{-3 x \left (\left (1-e^{-x}\right ) x^2+1\right )} xdx-3 \int e^{-3 x \left (\left (1-e^{-x}\right ) x^2+1\right )} x^2dx-3 \int e^{-3 \left (\left (1-e^{-x}\right ) x^2+1\right ) x-x} x^5dx-9 \int e^{-3 x \left (\left (1-e^{-x}\right ) x^2+1\right )} x^4dx+9 \int e^{-3 \left (\left (1-e^{-x}\right ) x^2+1\right ) x-x} x^4dx+x^3\) |
Input:
Int[(3*E^x*x^2 + E^((3*x^3 + E^x*(-3*x - 3*x^3))/E^x)*(9*x^4 - 3*x^5 + E^x *(2*x - 3*x^2 - 9*x^4)))/E^x,x]
Output:
$Aborted
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x^{3}+x^{2} {\mathrm e}^{-3 x \left ({\mathrm e}^{x} x^{2}-x^{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}\) | \(31\) |
parallelrisch | \(x^{3}+x^{2} {\mathrm e}^{\left (\left (-3 x^{3}-3 x \right ) {\mathrm e}^{x}+3 x^{3}\right ) {\mathrm e}^{-x}}\) | \(33\) |
norman | \(\left ({\mathrm e}^{x} x^{3}+{\mathrm e}^{x} x^{2} {\mathrm e}^{\left (\left (-3 x^{3}-3 x \right ) {\mathrm e}^{x}+3 x^{3}\right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\) | \(43\) |
Input:
int((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^ 3)/exp(x))+3*exp(x)*x^2)/exp(x),x,method=_RETURNVERBOSE)
Output:
x^3+x^2*exp(-3*x*(exp(x)*x^2-x^2+exp(x))*exp(-x))
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 \, {\left (x^{3} - {\left (x^{3} + x\right )} e^{x}\right )} e^{\left (-x\right )}\right )} \] Input:
integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x )+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x),x, algorithm="fricas")
Output:
x^3 + x^2*e^(3*(x^3 - (x^3 + x)*e^x)*e^(-x))
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 x^{3} + \left (- 3 x^{3} - 3 x\right ) e^{x}\right ) e^{- x}} \] Input:
integrate((((-9*x**4-3*x**2+2*x)*exp(x)-3*x**5+9*x**4)*exp(((-3*x**3-3*x)* exp(x)+3*x**3)/exp(x))+3*exp(x)*x**2)/exp(x),x)
Output:
x**3 + x**2*exp((3*x**3 + (-3*x**3 - 3*x)*exp(x))*exp(-x))
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^{3} + x^{2} e^{\left (3 \, x^{3} e^{\left (-x\right )} - 3 \, x^{3} - 3 \, x\right )} \] Input:
integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x )+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x),x, algorithm="maxima")
Output:
x^3 + x^2*e^(3*x^3*e^(-x) - 3*x^3 - 3*x)
\[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=\int { {\left (3 \, x^{2} e^{x} - {\left (3 \, x^{5} - 9 \, x^{4} + {\left (9 \, x^{4} + 3 \, x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (3 \, {\left (x^{3} - {\left (x^{3} + x\right )} e^{x}\right )} e^{\left (-x\right )}\right )}\right )} e^{\left (-x\right )} \,d x } \] Input:
integrate((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x )+3*x^3)/exp(x))+3*exp(x)*x^2)/exp(x),x, algorithm="giac")
Output:
integrate((3*x^2*e^x - (3*x^5 - 9*x^4 + (9*x^4 + 3*x^2 - 2*x)*e^x)*e^(3*(x ^3 - (x^3 + x)*e^x)*e^(-x)))*e^(-x), x)
Time = 6.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=x^3+x^2\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-3\,x^3}\,{\mathrm {e}}^{3\,x^3\,{\mathrm {e}}^{-x}} \] Input:
int(exp(-x)*(3*x^2*exp(x) - exp(-exp(-x)*(exp(x)*(3*x + 3*x^3) - 3*x^3))*( 3*x^5 - 9*x^4 + exp(x)*(3*x^2 - 2*x + 9*x^4))),x)
Output:
x^3 + x^2*exp(-3*x)*exp(-3*x^3)*exp(3*x^3*exp(-x))
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int e^{-x} \left (3 e^x x^2+e^{e^{-x} \left (3 x^3+e^x \left (-3 x-3 x^3\right )\right )} \left (9 x^4-3 x^5+e^x \left (2 x-3 x^2-9 x^4\right )\right )\right ) \, dx=\frac {x^{2} \left (e^{3 x^{3}+3 x} x +e^{\frac {3 x^{3}}{e^{x}}}\right )}{e^{3 x^{3}+3 x}} \] Input:
int((((-9*x^4-3*x^2+2*x)*exp(x)-3*x^5+9*x^4)*exp(((-3*x^3-3*x)*exp(x)+3*x^ 3)/exp(x))+3*exp(x)*x^2)/exp(x),x)
Output:
(x**2*(e**(3*x**3 + 3*x)*x + e**((3*x**3)/e**x)))/e**(3*x**3 + 3*x)