\(\int e^{-2 e^x} (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} (4 x+4 x^2-4 e^x x^2+4 x^3)) \, dx\) [5941]

Optimal result

Integrand size = 61, antiderivative size = 24 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 e^{2 x+x^2} \left (10+e^{-2 e^x} x^2\right ) \]

[Out]

2*exp(x^2+2*x)*(10+x^2/exp(exp(x))^2)

Rubi [F]

\[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=\int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx \]

[In]

Int[(E^(2*E^x + 2*x + x^2)*(40 + 40*x) + E^(2*x + x^2)*(4*x + 4*x^2 - 4*E^x*x^2 + 4*x^3))/E^(2*E^x),x]

[Out]

20*E^(2*x + x^2) + 4*Defer[Int][E^(-2*E^x + x*(2 + x))*x, x] + 4*Defer[Int][E^(-2*E^x + x*(2 + x))*x^2, x] - 4
*Defer[Int][E^(-2*E^x + x + x*(2 + x))*x^2, x] + 4*Defer[Int][E^(-2*E^x + x*(2 + x))*x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int e^{-2 e^x+x (2+x)} \left (-4 e^x x^2+40 e^{2 e^x} (1+x)+4 x \left (1+x+x^2\right )\right ) \, dx \\ & = \int \left (-4 e^{-2 e^x+x+x (2+x)} x^2+40 e^{x (2+x)} (1+x)+4 e^{-2 e^x+x (2+x)} x \left (1+x+x^2\right )\right ) \, dx \\ & = -\left (4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx\right )+4 \int e^{-2 e^x+x (2+x)} x \left (1+x+x^2\right ) \, dx+40 \int e^{x (2+x)} (1+x) \, dx \\ & = -\left (4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx\right )+4 \int \left (e^{-2 e^x+x (2+x)} x+e^{-2 e^x+x (2+x)} x^2+e^{-2 e^x+x (2+x)} x^3\right ) \, dx+40 \int e^{2 x+x^2} (1+x) \, dx \\ & = 20 e^{2 x+x^2}+4 \int e^{-2 e^x+x (2+x)} x \, dx+4 \int e^{-2 e^x+x (2+x)} x^2 \, dx-4 \int e^{-2 e^x+x+x (2+x)} x^2 \, dx+4 \int e^{-2 e^x+x (2+x)} x^3 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 e^{-2 e^x+x (2+x)} \left (10 e^{2 e^x}+x^2\right ) \]

[In]

Integrate[(E^(2*E^x + 2*x + x^2)*(40 + 40*x) + E^(2*x + x^2)*(4*x + 4*x^2 - 4*E^x*x^2 + 4*x^3))/E^(2*E^x),x]

[Out]

2*E^(-2*E^x + x*(2 + x))*(10*E^(2*E^x) + x^2)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

method	result	size
risch	\(20 \,{\mathrm e}^{x \left (2+x \right )}+2 x^{2} {\mathrm e}^{x^{2}-2 \,{\mathrm e}^{x}+2 x}\)	\(27\)
parallelrisch	\(\left (2 \,{\mathrm e}^{x^{2}+2 x} x^{2}+20 \,{\mathrm e}^{x^{2}+2 x} {\mathrm e}^{2 \,{\mathrm e}^{x}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{x}}\)	\(36\)

[In]

int(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,method
=_RETURNVERBOSE)

[Out]

20*exp(x*(2+x))+2*x^2*exp(x^2-2*exp(x)+2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 \, {\left (x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + 10 \, e^{\left (2 \, x^{2} + 4 \, x + 2 \, e^{x}\right )}\right )} e^{\left (-x^{2} - 2 \, x - 2 \, e^{x}\right )} \]

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="fricas")

[Out]

2*(x^2*e^(2*x^2 + 4*x) + 10*e^(2*x^2 + 4*x + 2*e^x))*e^(-x^2 - 2*x - 2*e^x)

Sympy [A] (verification not implemented)

Time = 16.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 x^{2} e^{x^{2} + 2 x} e^{- 2 e^{x}} + 20 e^{x^{2} + 2 x} \]

[In]

integrate(((40*x+40)*exp(x**2+2*x)*exp(exp(x))**2+(-4*exp(x)*x**2+4*x**3+4*x**2+4*x)*exp(x**2+2*x))/exp(exp(x)
)**2,x)

[Out]

2*x**2*exp(x**2 + 2*x)*exp(-2*exp(x)) + 20*exp(x**2 + 2*x)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.00 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 \, x^{2} e^{\left (x^{2} + 2 \, x - 2 \, e^{x}\right )} - 20 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + i\right ) e^{\left (-1\right )} - 20 \, {\left (\frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} - e^{\left ({\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} \]

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="maxima")

[Out]

2*x^2*e^(x^2 + 2*x - 2*e^x) - 20*I*sqrt(pi)*erf(I*x + I)*e^(-1) - 20*(sqrt(pi)*(x + 1)*(erf(sqrt(-(x + 1)^2))
- 1)/sqrt(-(x + 1)^2) - e^((x + 1)^2))*e^(-1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx=2 \, x^{2} e^{\left (x^{2} + 2 \, x - 2 \, e^{x}\right )} + 20 \, e^{\left (x^{2} + 2 \, x\right )} \]

[In]

integrate(((40*x+40)*exp(x^2+2*x)*exp(exp(x))^2+(-4*exp(x)*x^2+4*x^3+4*x^2+4*x)*exp(x^2+2*x))/exp(exp(x))^2,x,
algorithm="giac")

[Out]

2*x^2*e^(x^2 + 2*x - 2*e^x) + 20*e^(x^2 + 2*x)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int e^{-2 e^x} \left (e^{2 e^x+2 x+x^2} (40+40 x)+e^{2 x+x^2} \left (4 x+4 x^2-4 e^x x^2+4 x^3\right )\right ) \, dx={\mathrm {e}}^{x^2+2\,x}\,\left (2\,x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}+20\right ) \]

[In]

int(exp(-2*exp(x))*(exp(2*x + x^2)*(4*x - 4*x^2*exp(x) + 4*x^2 + 4*x^3) + exp(2*x + x^2)*exp(2*exp(x))*(40*x +
40)),x)

[Out]

exp(2*x + x^2)*(2*x^2*exp(-2*exp(x)) + 20)