3.5.70 \(\int \frac {e^{-x} (e^{2 x} (-1+x)+e^{e^{-x} x} (4 x^2-4 x^3))}{4 x^2} \, dx\)

Optimal. Leaf size=22 \[ e^{e^{-x} x}+\frac {e^x}{4 x}+\log (3) \]

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Rubi [F]  time = 0.73, 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 \frac {e^{-x} \left (e^{2 x} (-1+x)+e^{e^{-x} x} \left (4 x^2-4 x^3\right )\right )}{4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{-x} \left (e^{2 x} (-1+x)+e^{e^{-x} x} \left (4 x^2-4 x^3\right )\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-x} (1-x) \left (-e^{2 x}+4 e^{e^{-x} x} x^2\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-4 e^{-x+e^{-x} x} (-1+x)+\frac {e^x (-1+x)}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^x (-1+x)}{x^2} \, dx-\int e^{-x+e^{-x} x} (-1+x) \, dx\\ &=\frac {e^x}{4 x}-\int e^{-e^{-x} \left (-1+e^x\right ) x} (-1+x) \, dx\\ &=\frac {e^x}{4 x}-\int \left (-e^{-e^{-x} \left (-1+e^x\right ) x}+e^{-e^{-x} \left (-1+e^x\right ) x} x\right ) \, dx\\ &=\frac {e^x}{4 x}+\int e^{-e^{-x} \left (-1+e^x\right ) x} \, dx-\int e^{-e^{-x} \left (-1+e^x\right ) x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 23, normalized size = 1.05 \begin {gather*} \frac {1}{4} \left (4 e^{e^{-x} x}+\frac {e^x}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*E^(x/E^x) + E^x/x)/4

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fricas [A]  time = 0.55, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (4 \, x e^{\left (x e^{\left (-x\right )} - x\right )} + 1\right )} e^{x}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x*e^(x*e^(-x) - x) + 1)*e^x/x

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giac [A]  time = 0.37, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (4 \, x e^{\left (x e^{\left (-x\right )} - x\right )} + 1\right )} e^{x}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x*e^(x*e^(-x) - x) + 1)*e^x/x

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maple [A]  time = 0.05, size = 16, normalized size = 0.73

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{4 x}+{\mathrm e}^{x \,{\mathrm e}^{-x}}\)	\(16\)
norman	\(\frac {\left ({\mathrm e}^{x} x \,{\mathrm e}^{x \,{\mathrm e}^{-x}}+\frac {{\mathrm e}^{2 x}}{4}\right ) {\mathrm e}^{-x}}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(x)/x+exp(x*exp(-x))

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maxima [C]  time = 0.69, size = 19, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, {\rm Ei}\relax (x) + e^{\left (x e^{\left (-x\right )}\right )} - \frac {1}{4} \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Ei(x) + e^(x*e^(-x)) - 1/4*gamma(-1, -x)

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mupad [B]  time = 0.49, size = 15, normalized size = 0.68 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}+\frac {{\mathrm {e}}^x}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*exp(-x)) + exp(x)/(4*x)

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sympy [A]  time = 0.20, size = 12, normalized size = 0.55 \begin {gather*} e^{x e^{- x}} + \frac {e^{x}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*exp(-x)) + exp(x)/(4*x)

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