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

Optimal. Leaf size=37 \[ -4 e^{-e^{-3-x+e^{-x} x}}-x-\frac {4-e^x+x}{x} \]

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Rubi [F]  time = 1.89, 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^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.67, size = 36, normalized size = 0.97 \begin {gather*} -4 e^{-e^{-3-x+e^{-x} x}}-\frac {4}{x}+\frac {e^x}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4/E^E^(-3 - x + x/E^x) - 4/x + E^x/x - x

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fricas [A]  time = 0.57, size = 57, normalized size = 1.54 \begin {gather*} -\frac {{\left ({\left (x^{2} - e^{x} + 4\right )} e^{\left (e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 4 \, x\right )} e^{\left (-e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x^2 - e^x + 4)*e^(e^(-((x + 3)*e^x - x)*e^(-x))) + 4*x)*e^(-e^(-((x + 3)*e^x - x)*e^(-x)))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (4 \, {\left (x^{3} + x^{2} e^{x} - x^{2}\right )} e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} - {\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - {\left (x^{2} - 4\right )} e^{x}\right )} e^{\left (e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x - e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 40, normalized size = 1.08

method	result	size
risch	\(-x -\frac {4}{x}+\frac {{\mathrm e}^{x}}{x}-4 \,{\mathrm e}^{-{\mathrm e}^{-\left ({\mathrm e}^{x} x +3 \,{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x-4/x+exp(x)/x-4*exp(-exp(-(exp(x)*x+3*exp(x)-x)*exp(-x)))

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maxima [C]  time = 0.72, size = 35, normalized size = 0.95 \begin {gather*} -x - \frac {4}{x} + {\rm Ei}\relax (x) - 4 \, e^{\left (-e^{\left (x e^{\left (-x\right )} - x - 3\right )}\right )} - \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 4/x + Ei(x) - 4*e^(-e^(x*e^(-x) - x - 3)) - gamma(-1, -x)

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mupad [B]  time = 1.13, size = 33, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^x}{x}-4\,{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}}-x-\frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/x - 4*exp(-exp(-x)*exp(-3)*exp(x*exp(-x))) - x - 4/x

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sympy [A]  time = 0.56, size = 27, normalized size = 0.73 \begin {gather*} - x - 4 e^{- e^{\left (x + \left (- x - 3\right ) e^{x}\right ) e^{- x}}} + \frac {e^{x}}{x} - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 4*exp(-exp((x + (-x - 3)*exp(x))*exp(-x))) + exp(x)/x - 4/x

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