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

Optimal. Leaf size=12 \[ e^{-1+e^{\frac {1}{x}}-x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6706} \begin {gather*} e^{-x+e^{\frac {1}{x}}-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 + E^x^(-1) - 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^{-1+e^{\frac {1}{x}}-x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 12, normalized size = 1.00 \begin {gather*} e^{-1+e^{\frac {1}{x}}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 + E^x^(-1) - x)

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 18, normalized size = 1.50 \begin {gather*} \frac {e^{\left (-x + e^{\frac {1}{x}} + \log \left (x^{2}\right ) - 1\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1/x)-x^2)*exp(log(x^2)+exp(1/x)-x-1)/x^4,x, algorithm="fricas")

[Out]

e^(-x + e^(1/x) + log(x^2) - 1)/x^2

________________________________________________________________________________________

giac [A]  time = 0.14, size = 10, normalized size = 0.83 \begin {gather*} e^{\left (-x + e^{\frac {1}{x}} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1/x)-x^2)*exp(log(x^2)+exp(1/x)-x-1)/x^4,x, algorithm="giac")

[Out]

e^(-x + e^(1/x) - 1)

________________________________________________________________________________________

maple [C]  time = 1.28, size = 60, normalized size = 5.00

method	result	size
risch	\({\mathrm e}^{-1-\frac {i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\frac {i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+{\mathrm e}^{\frac {1}{x}}-x}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(1/x)-x^2)*exp(ln(x^2)+exp(1/x)-x-1)/x^4,x,method=_RETURNVERBOSE)

[Out]

exp(-1-1/2*I*Pi*csgn(I*x^2)^3+I*Pi*csgn(I*x)*csgn(I*x^2)^2-1/2*I*Pi*csgn(I*x)^2*csgn(I*x^2)+exp(1/x)-x)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 10, normalized size = 0.83 \begin {gather*} e^{\left (-x + e^{\frac {1}{x}} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1/x)-x^2)*exp(log(x^2)+exp(1/x)-x-1)/x^4,x, algorithm="maxima")

[Out]

e^(-x + e^(1/x) - 1)

________________________________________________________________________________________

mupad [B]  time = 4.32, size = 12, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{{\mathrm {e}}^{1/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(1/x) - x + log(x^2) - 1)*(exp(1/x) + x^2))/x^4,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 8, normalized size = 0.67 \begin {gather*} e^{- x + e^{\frac {1}{x}} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(1/x)-x**2)*exp(ln(x**2)+exp(1/x)-x-1)/x**4,x)

[Out]

exp(-x + exp(1/x) - 1)

________________________________________________________________________________________