3.33.71 \(\int e^{-2 x-4 e^{e^x} x} (1+e^{2 x+4 e^{e^x} x}-2 x+e^{e^x} (-4 x-4 e^x x^2)) \, dx\)

Optimal. Leaf size=18 \[ x+e^{-2 x-4 e^{e^x} x} x \]

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Rubi [F]  time = 2.02, 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^{-2 x-4 e^{e^x} x} \left (1+e^{2 x+4 e^{e^x} x}-2 x+e^{e^x} \left (-4 x-4 e^x x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.65, size = 18, normalized size = 1.00 \begin {gather*} x+e^{-2 x-4 e^{e^x} x} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + E^(-2*x - 4*E^E^x*x)*x

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fricas [A]  time = 0.57, size = 27, normalized size = 1.50 \begin {gather*} {\left (x e^{\left (4 \, x e^{\left (e^{x}\right )} + 2 \, x\right )} + x\right )} e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x*exp(exp(x))+x)^2+(-4*exp(x)*x^2-4*x)*exp(exp(x))+1-2*x)/exp(2*x*exp(exp(x))+x)^2,x, algorit
hm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (4 \, {\left (x^{2} e^{x} + x\right )} e^{\left (e^{x}\right )} + 2 \, x - e^{\left (4 \, x e^{\left (e^{x}\right )} + 2 \, x\right )} - 1\right )} e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x*exp(exp(x))+x)^2+(-4*exp(x)*x^2-4*x)*exp(exp(x))+1-2*x)/exp(2*x*exp(exp(x))+x)^2,x, algorit
hm="giac")

[Out]

integrate(-(4*(x^2*e^x + x)*e^(e^x) + 2*x - e^(4*x*e^(e^x) + 2*x) - 1)*e^(-4*x*e^(e^x) - 2*x), x)

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maple [A]  time = 0.04, size = 16, normalized size = 0.89

method	result	size
risch	\(x +x \,{\mathrm e}^{-2 x \left (2 \,{\mathrm e}^{{\mathrm e}^{x}}+1\right )}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x*exp(exp(x))+x)^2+(-4*exp(x)*x^2-4*x)*exp(exp(x))+1-2*x)/exp(2*x*exp(exp(x))+x)^2,x,method=_RETURN
VERBOSE)

[Out]

x+x*exp(-2*x*(2*exp(exp(x))+1))

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maxima [A]  time = 0.51, size = 15, normalized size = 0.83 \begin {gather*} x e^{\left (-4 \, x e^{\left (e^{x}\right )} - 2 \, x\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x*exp(exp(x))+x)^2+(-4*exp(x)*x^2-4*x)*exp(exp(x))+1-2*x)/exp(2*x*exp(exp(x))+x)^2,x, algorit
hm="maxima")

[Out]

x*e^(-4*x*e^(e^x) - 2*x) + x

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mupad [B]  time = 2.09, size = 15, normalized size = 0.83 \begin {gather*} x+x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + x*exp(-2*x)*exp(-4*x*exp(exp(x)))

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sympy [A]  time = 4.09, size = 17, normalized size = 0.94 \begin {gather*} x e^{- 4 x e^{e^{x}} - 2 x} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(-4*x*exp(exp(x)) - 2*x) + x

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