\(\int \frac {e^{\frac {1+11 x-x^2}{x}} (-1-x^2)}{x^2} \, dx\) [8301]

Optimal result

Integrand size = 27, antiderivative size = 10 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{11+\frac {1}{x}-x} \]

[Out]

exp(11+1/x-x)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6820, 6838} \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{-x+\frac {1}{x}+11} \]

[In]

Int[(E^((1 + 11*x - x^2)/x)*(-1 - x^2))/x^2,x]

[Out]

E^(11 + x^(-1) - 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^{11+\frac {1}{x}-x} \left (-1-x^2\right )}{x^2} \, dx \\ & = e^{11+\frac {1}{x}-x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{11+\frac {1}{x}-x} \]

[In]

Integrate[(E^((1 + 11*x - x^2)/x)*(-1 - x^2))/x^2,x]

[Out]

E^(11 + x^(-1) - x)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.50

method	result	size
gosper	\({\mathrm e}^{-\frac {x^{2}-11 x -1}{x}}\)	\(15\)
risch	\({\mathrm e}^{-\frac {x^{2}-11 x -1}{x}}\)	\(15\)
parallelrisch	\({\mathrm e}^{-\frac {x^{2}-11 x -1}{x}}\)	\(15\)
norman	\({\mathrm e}^{\frac {-x^{2}+11 x +1}{x}}\)	\(16\)

[In]

int((-x^2-1)*exp((-x^2+11*x+1)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-(x^2-11*x-1)/x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{\left (-\frac {x^{2} - 11 \, x - 1}{x}\right )} \]

[In]

integrate((-x^2-1)*exp((-x^2+11*x+1)/x)/x^2,x, algorithm="fricas")

[Out]

e^(-(x^2 - 11*x - 1)/x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{\frac {- x^{2} + 11 x + 1}{x}} \]

[In]

integrate((-x**2-1)*exp((-x**2+11*x+1)/x)/x**2,x)

[Out]

exp((-x**2 + 11*x + 1)/x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{\left (-x + \frac {1}{x} + 11\right )} \]

[In]

integrate((-x^2-1)*exp((-x^2+11*x+1)/x)/x^2,x, algorithm="maxima")

[Out]

e^(-x + 1/x + 11)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx=e^{\left (-x + \frac {1}{x} + 11\right )} \]

[In]

integrate((-x^2-1)*exp((-x^2+11*x+1)/x)/x^2,x, algorithm="giac")

[Out]

e^(-x + 1/x + 11)

Mupad [B] (verification not implemented)

Time = 15.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {1+11 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx={\mathrm {e}}^{-x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{11} \]

[In]

int(-(exp((11*x - x^2 + 1)/x)*(x^2 + 1))/x^2,x)

[Out]

exp(-x)*exp(1/x)*exp(11)