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

Optimal. Leaf size=21 \[ -2 \left (e^{4-e^{2 x} x}+x\right )-10 \log (4) \]

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Rubi [F]  time = 0.14, 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 \left (-2+e^{4+2 x-e^{2 x} x} (2+4 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 18, normalized size = 0.86 \begin {gather*} -2 e^{4-e^{2 x} x}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*E^(4 - E^(2*x)*x) - 2*x

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fricas [A]  time = 0.54, size = 26, normalized size = 1.24 \begin {gather*} -2 \, {\left (x e^{\left (2 \, x\right )} + e^{\left (-x e^{\left (2 \, x\right )} + 2 \, x + 4\right )}\right )} e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.28, size = 16, normalized size = 0.76 \begin {gather*} -2 \, x - 2 \, e^{\left (-x e^{\left (2 \, x\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x - 2*e^(-x*e^(2*x) + 4)

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maple [A]  time = 0.06, size = 17, normalized size = 0.81

method	result	size
default	\(-2 x -2 \,{\mathrm e}^{-x \,{\mathrm e}^{2 x}+4}\)	\(17\)
norman	\(-2 x -2 \,{\mathrm e}^{-x \,{\mathrm e}^{2 x}+4}\)	\(17\)
risch	\(-2 x -2 \,{\mathrm e}^{-x \,{\mathrm e}^{2 x}+4}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.39, size = 16, normalized size = 0.76 \begin {gather*} -2 \, x - 2 \, e^{\left (-x e^{\left (2 \, x\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x - 2*e^(-x*e^(2*x) + 4)

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mupad [B]  time = 4.21, size = 16, normalized size = 0.76 \begin {gather*} -2\,x-2\,{\mathrm {e}}^4\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2*x - 2*exp(4)*exp(-x*exp(2*x))

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sympy [A]  time = 0.15, size = 15, normalized size = 0.71 \begin {gather*} - 2 x - 2 e^{- x e^{2 x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x - 2*exp(-x*exp(2*x) + 4)

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