Integrand size = 93, antiderivative size = 30 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=e^{-e^{x^2}-2 x} x (e+x) \left (5+\frac {1}{2} (2-x) x\right ) \] Output:
x*(x+exp(1))/exp(exp(x^2)+x)/exp(x)*(1/2*(2-x)*x+5)
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=-\frac {1}{2} e^{-e^{x^2}-2 x} x (e+x) \left (-10-2 x+x^2\right ) \] Input:
Integrate[(E^(-E^x^2 - 2*x)*(20*x - 14*x^2 - 8*x^3 + 2*x^4 + E*(10 - 16*x - 7*x^2 + 2*x^3) + E^x^2*(-20*x^3 - 4*x^4 + 2*x^5 + E*(-20*x^2 - 4*x^3 + 2 *x^4))))/2,x]
Output:
-1/2*(E^(-E^x^2 - 2*x)*x*(E + x)*(-10 - 2*x + x^2))
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.
\(\displaystyle \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (2 x^4-8 x^3-14 x^2+e \left (2 x^3-7 x^2-16 x+10\right )+e^{x^2} \left (2 x^5-4 x^4-20 x^3+e \left (2 x^4-4 x^3-20 x^2\right )\right )+20 x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int e^{-2 x-e^{x^2}} \left (2 x^4-8 x^3-14 x^2+20 x+e \left (2 x^3-7 x^2-16 x+10\right )-2 e^{x^2} \left (-x^5+2 x^4+10 x^3+e \left (-x^4+2 x^3+10 x^2\right )\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (2 e^{-2 x-e^{x^2}} x^4-8 e^{-2 x-e^{x^2}} x^3-14 e^{-2 x-e^{x^2}} x^2+2 e^{x^2-2 x-e^{x^2}} (x+e) \left (x^2-2 x-10\right ) x^2+20 e^{-2 x-e^{x^2}} x+e^{-2 x-e^{x^2}+1} \left (2 x^3-7 x^2-16 x+10\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (10 \int e^{-2 x-e^{x^2}+1}dx+20 \int e^{-2 x-e^{x^2}} xdx-16 \int e^{-2 x-e^{x^2}+1} xdx-14 \int e^{-2 x-e^{x^2}} x^2dx-7 \int e^{-2 x-e^{x^2}+1} x^2dx-20 \int e^{x^2-2 x-e^{x^2}+1} x^2dx+2 \int e^{x^2-2 x-e^{x^2}} x^5dx+2 \int e^{-2 x-e^{x^2}} x^4dx-2 (2-e) \int e^{x^2-2 x-e^{x^2}} x^4dx-8 \int e^{-2 x-e^{x^2}} x^3dx+2 \int e^{-2 x-e^{x^2}+1} x^3dx-4 (5+e) \int e^{x^2-2 x-e^{x^2}} x^3dx\right )\) |
Input:
Int[(E^(-E^x^2 - 2*x)*(20*x - 14*x^2 - 8*x^3 + 2*x^4 + E*(10 - 16*x - 7*x^ 2 + 2*x^3) + E^x^2*(-20*x^3 - 4*x^4 + 2*x^5 + E*(-20*x^2 - 4*x^3 + 2*x^4)) ))/2,x]
Output:
$Aborted
Time = 1.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40
method | result | size |
risch | \(-\frac {\left (x^{2} {\mathrm e}+x^{3}-2 x \,{\mathrm e}-2 x^{2}-10 \,{\mathrm e}-10 x \right ) x \,{\mathrm e}^{-2 x -{\mathrm e}^{x^{2}}}}{2}\) | \(42\) |
parallelrisch | \(-\frac {\left (x^{3} {\mathrm e}+x^{4}-2 x^{2} {\mathrm e}-2 x^{3}-10 x \,{\mathrm e}-10 x^{2}\right ) {\mathrm e}^{-x} {\mathrm e}^{-{\mathrm e}^{x^{2}}-x}}{2}\) | \(48\) |
Input:
int(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3- 7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x,me thod=_RETURNVERBOSE)
Output:
-1/2*(x^2*exp(1)+x^3-2*x*exp(1)-2*x^2-10*exp(1)-10*x)*x*exp(-2*x-exp(x^2))
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=-\frac {1}{2} \, {\left (x^{4} - 2 \, x^{3} - 10 \, x^{2} + {\left (x^{3} - 2 \, x^{2} - 10 \, x\right )} e\right )} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} \] Input:
integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+( 2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x ),x, algorithm="fricas")
Output:
-1/2*(x^4 - 2*x^3 - 10*x^2 + (x^3 - 2*x^2 - 10*x)*e)*e^(-2*x - e^(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
Time = 45.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=\frac {\left (- x^{4} e^{- x} - e x^{3} e^{- x} + 2 x^{3} e^{- x} + 2 e x^{2} e^{- x} + 10 x^{2} e^{- x} + 10 e x e^{- x}\right ) e^{- x - e^{x^{2}}}}{2} \] Input:
integrate(1/2*(((2*x**4-4*x**3-20*x**2)*exp(1)+2*x**5-4*x**4-20*x**3)*exp( x**2)+(2*x**3-7*x**2-16*x+10)*exp(1)+2*x**4-8*x**3-14*x**2+20*x)/exp(x)/ex p(exp(x**2)+x),x)
Output:
(-x**4*exp(-x) - E*x**3*exp(-x) + 2*x**3*exp(-x) + 2*E*x**2*exp(-x) + 10*x **2*exp(-x) + 10*E*x*exp(-x))*exp(-x - exp(x**2))/2
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=-\frac {1}{2} \, {\left (x^{4} + x^{3} {\left (e - 2\right )} - 2 \, x^{2} {\left (e + 5\right )} - 10 \, x e\right )} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} \] Input:
integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+( 2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x ),x, algorithm="maxima")
Output:
-1/2*(x^4 + x^3*(e - 2) - 2*x^2*(e + 5) - 10*x*e)*e^(-2*x - e^(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=-\frac {1}{2} \, x^{4} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} - \frac {1}{2} \, x^{3} e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} + x^{3} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} + x^{2} e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} + 5 \, x^{2} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} + 5 \, x e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} \] Input:
integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+( 2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x ),x, algorithm="giac")
Output:
-1/2*x^4*e^(-2*x - e^(x^2)) - 1/2*x^3*e^(-2*x - e^(x^2) + 1) + x^3*e^(-2*x - e^(x^2)) + x^2*e^(-2*x - e^(x^2) + 1) + 5*x^2*e^(-2*x - e^(x^2)) + 5*x* e^(-2*x - e^(x^2) + 1)
Time = 1.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-2\,x-{\mathrm {e}}^{x^2}}\,\left (x+\mathrm {e}\right )\,\left (-x^2+2\,x+10\right )}{2} \] Input:
int(-exp(- x - exp(x^2))*exp(-x)*((exp(x^2)*(exp(1)*(20*x^2 + 4*x^3 - 2*x^ 4) + 20*x^3 + 4*x^4 - 2*x^5))/2 - 10*x + (exp(1)*(16*x + 7*x^2 - 2*x^3 - 1 0))/2 + 7*x^2 + 4*x^3 - x^4),x)
Output:
(x*exp(- 2*x - exp(x^2))*(x + exp(1))*(2*x - x^2 + 10))/2
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx=\frac {x \left (-e \,x^{2}-x^{3}+2 e x +2 x^{2}+10 e +10 x \right )}{2 e^{e^{x^{2}}+2 x}} \] Input:
int(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3- 7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x)
Output:
(x*( - e*x**2 + 2*e*x + 10*e - x**3 + 2*x**2 + 10*x))/(2*e**(e**(x**2) + 2 *x))