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

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

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Rubi [F]  time = 0.81, 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^{e^{\frac {4+e^2 x-x^2}{x}} x+\frac {4+e^2 x-x^2}{x}} \left (-4+x-x^2\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.40, size = 18, normalized size = 1.00 \begin {gather*} e^{e^{e^2+\frac {4}{x}-x} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^(E^2 + 4/x - x)*x)

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fricas [B]  time = 0.56, size = 51, normalized size = 2.83 \begin {gather*} e^{\left (\frac {x^{2} e^{\left (-\frac {x^{2} - x e^{2} - 4}{x}\right )} - x^{2} + x e^{2} + 4}{x} + \frac {x^{2} - x e^{2} - 4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.19, size = 20, normalized size = 1.11

method	result	size
norman	\({\mathrm e}^{x \,{\mathrm e}^{\frac {{\mathrm e}^{2} x -x^{2}+4}{x}}}\)	\(20\)
risch	\({\mathrm e}^{x \,{\mathrm e}^{\frac {{\mathrm e}^{2} x -x^{2}+4}{x}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 15, normalized size = 0.83 \begin {gather*} e^{\left (x e^{\left (-x + \frac {4}{x} + e^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x*e^(-x + 4/x + e^2))

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mupad [B]  time = 3.67, size = 16, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{{\mathrm {e}}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.37, size = 15, normalized size = 0.83 \begin {gather*} e^{x e^{\frac {- x^{2} + x e^{2} + 4}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*exp((-x**2 + x*exp(2) + 4)/x))

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