3.2.67 \(\int \frac {e^{-x} (2+x)}{x^3} \, dx\) [167]

Optimal. Leaf size=10 \[ -\frac {e^{-x}}{x^2} \]

[Out]

-1/exp(x)/x^2

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2228} \begin {gather*} -\frac {e^{-x}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(E^x*x^3),x]

[Out]

-(1/(E^x*x^2))

Rule 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\frac {e^{-x}}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} -\frac {e^{-x}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(E^x*x^3),x]

[Out]

-(1/(E^x*x^2))

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Maple [A]
time = 0.04, size = 10, normalized size = 1.00

method	result	size
gosper	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
derivativedivides	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
default	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
norman	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
risch	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
parallelrisch	\(-\frac {{\mathrm e}^{-x}}{x^{2}}\)	\(10\)
meijerg	\(-\frac {1}{x^{2}}+\frac {1}{x}-\frac {1}{2}+\frac {9 x^{2}-12 x +6}{6 x^{2}}-\frac {\left (3-3 x \right ) {\mathrm e}^{-x}}{3 x^{2}}+\frac {-2 x +2}{2 x}-\frac {{\mathrm e}^{-x}}{x}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.35, size = 11, normalized size = 1.10 \begin {gather*} -\Gamma \left (-1, x\right ) - 2 \, \Gamma \left (-2, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(-1, x) - 2*gamma(-2, x)

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Fricas [A]
time = 0.55, size = 9, normalized size = 0.90 \begin {gather*} -\frac {e^{\left (-x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-x)/x^2

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Sympy [A]
time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} - \frac {e^{- x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-x)/x**2

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Giac [A]
time = 0.48, size = 9, normalized size = 0.90 \begin {gather*} -\frac {e^{\left (-x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-x)/x^2

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Mupad [B]
time = 0.05, size = 9, normalized size = 0.90 \begin {gather*} -\frac {{\mathrm {e}}^{-x}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(x + 2))/x^3,x)

[Out]

-exp(-x)/x^2

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Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {{\mathrm e}^{x} \left (x +4\right )}{x^{2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((2+x)*exp(-x)/x^3,x)

[Out]

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

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