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

Optimal. Leaf size=18 $$x+e^{-2x-4e^{e^x}x}x$$

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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 = {} $$\begin{array}{c}\int e^{-2x-4e^{e^x}x}(1+e^{2x+4e^{e^x}x}-2x+e^{e^x}(-4x-4e^x{x}^2))dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int e^{-2(1+2e^{e^x})x}(1+e^{2x+4e^{e^x}x}-2x+e^{e^x}(-4x-4e^x{x}^2))dx\\ & =\int (1+e^{-2(1+2e^{e^x})x}-2e^{-2(1+2e^{e^x})x}x-4e^{e^x-2(1+2e^{e^x})x}x(1+e^{x}x))dx\\ & =x-2\int e^{-2(1+2e^{e^x})x}xdx-4\int e^{e^x-2(1+2e^{e^x})x}x(1+e^{x}x)dx+\int e^{-2(1+2e^{e^x})x}dx\\ & =x-2\int e^{-2(1+2e^{e^x})x}xdx-4\int (e^{e^x-2(1+2e^{e^x})x}x+e^{e^x+x-2(1+2e^{e^x})x}{x}^2)dx+\int e^{-2(1+2e^{e^x})x}dx\\ & =x-2\int e^{-2(1+2e^{e^x})x}xdx-4\int e^{e^x-2(1+2e^{e^x})x}xdx-4\int e^{e^x+x-2(1+2e^{e^x})x}{x}^{2}dx+\int e^{-2(1+2e^{e^x})x}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}(x{e}^{(4x{e}^{\left(e^x\right)}+2x)}+x)e^{(-4x{e}^{\left(e^x\right)}-2x)}\end{array}$$

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 $$\begin{array}{c}\int -(4(x^2{e}^x+x)e^{\left(e^x\right)}+2x-e^{(4x{e}^{\left(e^x\right)}+2x)}-1)e^{(-4x{e}^{\left(e^x\right)}-2x)}dx\end{array}$$

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	$x+x{\mathrm{e}}^{-2x(2{\mathrm{e}}^{{\mathrm{e}}^x}+1)}$	$16$

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 $$\begin{array}{c}x{e}^{(-4x{e}^{\left(e^x\right)}-2x)}+x\end{array}$$

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 $$\begin{array}{c}x+x{\mathrm{e}}^{-2x}{\mathrm{e}}^{-4x{\mathrm{e}}^{{\mathrm{e}}^x}}\end{array}$$

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 $$\begin{array}{c}x{e}^{-4x{e}^{e^x}-2x}+x\end{array}$$

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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