\(\int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} (2+x^2)}{x} \, dx\) [1208]

Optimal result

Integrand size = 29, antiderivative size = 14 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-e^{12+\frac {2}{x}-x} \]

[Out]

-exp(2/x-x-ln(x))*x*exp(3)^4

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6820, 6838} \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-e^{-x+\frac {2}{x}+12} \]

[In]

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

[Out]

-E^(12 + 2/x - x)

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-e^{12+\frac {2}{x}-x} \]

[In]

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

[Out]

-E^(12 + 2/x - x)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-x e^{\left (-\frac {x^{2} + x \log \left (x\right ) - 12 \, x - 2}{x}\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=- x e^{12} e^{\frac {- x^{2} - x \log {\left (x \right )} + 2}{x}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-e^{\left (-x + \frac {2}{x} + 12\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-e^{\left (-\frac {x^{2} - 12 \, x - 2}{x}\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 8.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {e^{12+\frac {2-x^2-x \log (x)}{x}} \left (2+x^2\right )}{x} \, dx=-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{12}\,{\mathrm {e}}^{2/x} \]

[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)