3.48.63 \(\int \frac {e^{-4 x} (-2-4 x)}{x^3} \, dx\)

Optimal. Leaf size=9 \[ \frac {e^{-4 x}}{x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2197} \begin {gather*} \frac {e^{-4 x}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - 4*x)/(E^(4*x)*x^3),x]

[Out]

1/(E^(4*x)*x^2)

Rule 2197

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^{-4 x}}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {e^{-4 x}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 4*x)/(E^(4*x)*x^3),x]

[Out]

1/(E^(4*x)*x^2)

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fricas [A]  time = 0.70, size = 8, normalized size = 0.89 \begin {gather*} \frac {e^{\left (-4 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-2)*exp(-2*x)^2/x^3,x, algorithm="fricas")

[Out]

e^(-4*x)/x^2

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giac [A]  time = 0.16, size = 8, normalized size = 0.89 \begin {gather*} \frac {e^{\left (-4 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-2)*exp(-2*x)^2/x^3,x, algorithm="giac")

[Out]

e^(-4*x)/x^2

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maple [A]  time = 0.09, size = 9, normalized size = 1.00

method	result	size
risch	\(\frac {{\mathrm e}^{-4 x}}{x^{2}}\)	\(9\)
gosper	\(\frac {{\mathrm e}^{-4 x}}{x^{2}}\)	\(11\)
derivativedivides	\(\frac {{\mathrm e}^{-4 x}}{x^{2}}\)	\(11\)
default	\(\frac {{\mathrm e}^{-4 x}}{x^{2}}\)	\(11\)
norman	\(\frac {{\mathrm e}^{-4 x}}{x^{2}}\)	\(11\)
meijerg	\(-\frac {2 \left (-8 x +2\right )}{x}+\frac {4 \,{\mathrm e}^{-4 x}}{x}+8-\frac {4}{x}-\frac {144 x^{2}-48 x +6}{6 x^{2}}+\frac {\left (-12 x +3\right ) {\mathrm e}^{-4 x}}{3 x^{2}}+\frac {1}{x^{2}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x-2)*exp(-2*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(-4*x)/x^2

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maxima [C]  time = 0.39, size = 15, normalized size = 1.67 \begin {gather*} 16 \, \Gamma \left (-1, 4 \, x\right ) + 32 \, \Gamma \left (-2, 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-2)*exp(-2*x)^2/x^3,x, algorithm="maxima")

[Out]

16*gamma(-1, 4*x) + 32*gamma(-2, 4*x)

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mupad [B]  time = 0.06, size = 8, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^{-4\,x}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4*x)*(4*x + 2))/x^3,x)

[Out]

exp(-4*x)/x^2

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sympy [A]  time = 0.08, size = 8, normalized size = 0.89 \begin {gather*} \frac {e^{- 4 x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-2)*exp(-2*x)**2/x**3,x)

[Out]

exp(-4*x)/x**2

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