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

Optimal. Leaf size=23 \[ e^{e^{-1-\frac {3}{e^3 x}}-e^x-x} \]

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Rubi [A]  time = 0.67, antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 1, number of rules used = 1, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} e^{-x-e^x+e^{-\frac {e^3 x+3}{e^3 x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{-e^x+e^{-\frac {3+e^3 x}{e^3 x}}-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.62, size = 23, normalized size = 1.00 \begin {gather*} e^{e^{-1-\frac {3}{e^3 x}}-e^x-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^(-1 - 3/(E^3*x)) - E^x - x)

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fricas [A]  time = 0.92, size = 36, normalized size = 1.57 \begin {gather*} e^{\left (-{\left ({\left (x + 3\right )} e^{3} + e^{\left (x + 3\right )} - e^{\left (-\frac {{\left (x e^{3} + 3\right )} e^{\left (-3\right )}}{x} + 3\right )}\right )} e^{\left (-3\right )} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 19, normalized size = 0.83 \begin {gather*} e^{\left (-x - e^{x} + e^{\left (-\frac {3 \, e^{\left (-3\right )}}{x} - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x - e^x + e^(-3*e^(-3)/x - 1))

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maple [A]  time = 0.14, size = 24, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-\frac {\left (x \,{\mathrm e}^{3}+3\right ) {\mathrm e}^{-3}}{x}}-x}\)	\(24\)
norman	\({\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{\frac {\left (-x \,{\mathrm e}^{3}-3\right ) {\mathrm e}^{-3}}{x}}-x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.13, size = 22, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^{-3}}{x}}\,{\mathrm {e}}^{-1}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(-(3*exp(-3))/x)*exp(-1))*exp(-x)*exp(-exp(x))

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sympy [A]  time = 0.40, size = 20, normalized size = 0.87 \begin {gather*} e^{- x - e^{x} + e^{\frac {- x e^{3} - 3}{x e^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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