3.13.8 \(\int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} (2+x^2)}{x} \, dx\)

Optimal. Leaf size=14 \[ -e^{12+\frac {2}{x}-x} \]

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Rubi [A]  time = 0.27, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6688, 6706} \begin {gather*} -e^{-x+\frac {2}{x}+12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(12 + (2 - x^2 - x*Log[x])/x)*(2 + x^2))/x,x]

[Out]

-E^(12 + 2/x - x)

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \frac {e^{12+\frac {2}{x}-x} \left (2+x^2\right )}{x^2} \, dx\\ &=-e^{12+\frac {2}{x}-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} -e^{12+\frac {2}{x}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(12 + (2 - x^2 - x*Log[x])/x)*(2 + x^2))/x,x]

[Out]

-E^(12 + 2/x - x)

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fricas [A]  time = 0.60, size = 21, normalized size = 1.50 \begin {gather*} -x e^{\left (-\frac {x^{2} + x \log \relax (x) - 12 \, x - 2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*e^(-(x^2 + x*log(x) - 12*x - 2)/x)

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giac [A]  time = 0.49, size = 16, normalized size = 1.14 \begin {gather*} -e^{\left (-\frac {x^{2} - 12 \, x - 2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-(x^2 - 12*x - 2)/x)

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maple [A]  time = 0.08, size = 17, normalized size = 1.21

method	result	size
risch	\(-{\mathrm e}^{-\frac {x^{2}-12 x -2}{x}}\)	\(17\)
gosper	\(-x \,{\mathrm e}^{12} {\mathrm e}^{-\frac {x \ln \relax (x )+x^{2}-2}{x}}\)	\(23\)
norman	\(-x \,{\mathrm e}^{12} {\mathrm e}^{\frac {-x \ln \relax (x )-x^{2}+2}{x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-(x^2-12*x-2)/x)

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maxima [A]  time = 0.64, size = 13, normalized size = 0.93 \begin {gather*} -e^{\left (-x + \frac {2}{x} + 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-x + 2/x + 12)

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mupad [B]  time = 0.86, size = 14, normalized size = 1.00 \begin {gather*} -{\mathrm {e}}^{-x}\,{\mathrm {e}}^{12}\,{\mathrm {e}}^{2/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(12)*exp(-(x*log(x) + x^2 - 2)/x)*(x^2 + 2))/x,x)

[Out]

-exp(-x)*exp(12)*exp(2/x)

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sympy [A]  time = 0.31, size = 19, normalized size = 1.36 \begin {gather*} - x e^{12} e^{\frac {- x^{2} - x \log {\relax (x )} + 2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(12)*exp((-x**2 - x*log(x) + 2)/x)

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