3.66.47 \(\int e^{-4 e^x} (-2 x-2 e^{4 e^x} x+4 e^x x^2) \, dx\)

Optimal. Leaf size=30 \[ i \pi -x \left (x+e^{-4 e^x} x\right )+\log \left (-2+e^{e^{4 e}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 e^{-4 e^x} x \left (-1-e^{4 e^x}+2 e^x x\right ) \, dx\\ &=2 \int e^{-4 e^x} x \left (-1-e^{4 e^x}+2 e^x x\right ) \, dx\\ &=2 \int \left (-e^{-4 e^x} \left (1+e^{4 e^x}\right ) x+2 e^{-4 e^x+x} x^2\right ) \, dx\\ &=-\left (2 \int e^{-4 e^x} \left (1+e^{4 e^x}\right ) x \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx\\ &=-\left (2 \int \left (1+e^{-4 e^x}\right ) x \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx\\ &=-\left (2 \int \left (x+e^{-4 e^x} x\right ) \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx\\ &=-x^2-2 \int e^{-4 e^x} x \, dx+4 \int e^{-4 e^x+x} x^2 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 14, normalized size = 0.47 \begin {gather*} -\left (\left (1+e^{-4 e^x}\right ) x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 + E^(-4*E^x))*x^2)

________________________________________________________________________________________

fricas [A]  time = 1.16, size = 20, normalized size = 0.67 \begin {gather*} -{\left (x^{2} e^{\left (4 \, e^{x}\right )} + x^{2}\right )} e^{\left (-4 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2*e^(4*e^x) + x^2)*e^(-4*e^x)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 24, normalized size = 0.80 \begin {gather*} -{\left (x^{2} e^{\left (x - 4 \, e^{x}\right )} + x^{2} e^{x}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2*e^(x - 4*e^x) + x^2*e^x)*e^(-x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 17, normalized size = 0.57

method	result	size
risch	\(-x^{2}-x^{2} {\mathrm e}^{-4 \,{\mathrm e}^{x}}\)	\(17\)
norman	\(\left (-x^{2}-x^{2} {\mathrm e}^{4 \,{\mathrm e}^{x}}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2-x^2*exp(-4*exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 16, normalized size = 0.53 \begin {gather*} -x^{2} e^{\left (-4 \, e^{x}\right )} - x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*e^(-4*e^x) - x^2

________________________________________________________________________________________

mupad [B]  time = 4.06, size = 12, normalized size = 0.40 \begin {gather*} -x^2\,\left ({\mathrm {e}}^{-4\,{\mathrm {e}}^x}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2*(exp(-4*exp(x)) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 14, normalized size = 0.47 \begin {gather*} - x^{2} - x^{2} e^{- 4 e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 - x**2*exp(-4*exp(x))

________________________________________________________________________________________