\(\int e^{-e^{x^2}} (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} (2+e^{6+x} (1+x))) \, dx\) [5112]

Optimal result

Integrand size = 45, antiderivative size = 22 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=x \left (2+e^{6+x}+e^{-e^{x^2}} x^2\right ) \]

[Out]

x*(2+exp(6+x)+x^2/exp(exp(x^2)))

Rubi [F]

\[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=\int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx \]

[In]

Int[(3*x^2 - 2*E^x^2*x^4 + E^E^x^2*(2 + E^(6 + x)*(1 + x)))/E^E^x^2,x]

[Out]

2*x + E^(6 + x)*x + 3*Defer[Int][x^2/E^E^x^2, x] - 2*Defer[Int][E^(-E^x^2 + x^2)*x^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (2+e^{6+x}+e^{6+x} x+3 e^{-e^{x^2}} x^2-2 e^{-e^{x^2}+x^2} x^4\right ) \, dx \\ & = 2 x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx+\int e^{6+x} \, dx+\int e^{6+x} x \, dx \\ & = e^{6+x}+2 x+e^{6+x} x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx-\int e^{6+x} \, dx \\ & = 2 x+e^{6+x} x-2 \int e^{-e^{x^2}+x^2} x^4 \, dx+3 \int e^{-e^{x^2}} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=2 x+e^{6+x} x+e^{-e^{x^2}} x^3 \]

[In]

Integrate[(3*x^2 - 2*E^x^2*x^4 + E^E^x^2*(2 + E^(6 + x)*(1 + x)))/E^E^x^2,x]

[Out]

2*x + E^(6 + x)*x + x^3/E^E^x^2

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

method	result	size
risch	\(x \,{\mathrm e}^{6+x}+2 x +x^{3} {\mathrm e}^{-{\mathrm e}^{x^{2}}}\)	\(22\)
parallelrisch	\(-\left (-x^{3}-{\mathrm e}^{6+x} {\mathrm e}^{{\mathrm e}^{x^{2}}} x -2 \,{\mathrm e}^{{\mathrm e}^{x^{2}}} x \right ) {\mathrm e}^{-{\mathrm e}^{x^{2}}}\)	\(36\)

[In]

int((((1+x)*exp(6+x)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x,method=_RETURNVERBOSE)

[Out]

x*exp(6+x)+2*x+x^3*exp(-exp(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx={\left (x^{3} + {\left (x e^{\left (x + 6\right )} + 2 \, x\right )} e^{\left (e^{\left (x^{2}\right )}\right )}\right )} e^{\left (-e^{\left (x^{2}\right )}\right )} \]

[In]

integrate((((1+x)*exp(6+x)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="fricas")

[Out]

(x^3 + (x*e^(x + 6) + 2*x)*e^(e^(x^2)))*e^(-e^(x^2))

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=x^{3} e^{- e^{x^{2}}} + x e^{x + 6} + 2 x \]

[In]

integrate((((1+x)*exp(6+x)+2)*exp(exp(x**2))-2*x**4*exp(x**2)+3*x**2)/exp(exp(x**2)),x)

[Out]

x**3*exp(-exp(x**2)) + x*exp(x + 6) + 2*x

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=x^{3} e^{\left (-e^{\left (x^{2}\right )}\right )} + {\left (x e^{6} - e^{6}\right )} e^{x} + 2 \, x + e^{\left (x + 6\right )} \]

[In]

integrate((((1+x)*exp(6+x)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="maxima")

[Out]

x^3*e^(-e^(x^2)) + (x*e^6 - e^6)*e^x + 2*x + e^(x + 6)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=x^{3} e^{\left (-e^{\left (x^{2}\right )}\right )} + x e^{\left (x + 6\right )} + 2 \, x \]

[In]

integrate((((1+x)*exp(6+x)+2)*exp(exp(x^2))-2*x^4*exp(x^2)+3*x^2)/exp(exp(x^2)),x, algorithm="giac")

[Out]

x^3*e^(-e^(x^2)) + x*e^(x + 6) + 2*x

Mupad [B] (verification not implemented)

Time = 13.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int e^{-e^{x^2}} \left (3 x^2-2 e^{x^2} x^4+e^{e^{x^2}} \left (2+e^{6+x} (1+x)\right )\right ) \, dx=2\,x+x^3\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}+x\,{\mathrm {e}}^6\,{\mathrm {e}}^x \]

[In]

int(exp(-exp(x^2))*(exp(exp(x^2))*(exp(x + 6)*(x + 1) + 2) - 2*x^4*exp(x^2) + 3*x^2),x)

[Out]

2*x + x^3*exp(-exp(x^2)) + x*exp(6)*exp(x)