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

Optimal. Leaf size=10 \[ e^{11+\frac {1}{x}-x} \]

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Rubi [A]  time = 0.21, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6688, 6706} \begin {gather*} e^{-x+\frac {1}{x}+11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.04, size = 10, normalized size = 1.00 \begin {gather*} e^{11+\frac {1}{x}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(11 + x^(-1) - x)

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fricas [A]  time = 0.66, size = 14, normalized size = 1.40 \begin {gather*} e^{\left (-\frac {x^{2} - 11 \, x - 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.13, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (-x + \frac {1}{x} + 11\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x + 1/x + 11)

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maple [A]  time = 0.11, size = 15, normalized size = 1.50

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.41, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (-x + \frac {1}{x} + 11\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-x + 1/x + 11)

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mupad [B]  time = 5.08, size = 11, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{-x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 10, normalized size = 1.00 \begin {gather*} e^{\frac {- x^{2} + 11 x + 1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((-x**2 + 11*x + 1)/x)

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