3.10.67 \(\int \frac {8-4 e}{e^4 x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac {8 \left (1-\frac {e}{2}\right )}{e^4 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 30} \begin {gather*} -\frac {4 (2-e)}{e^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 - 4*E)/(E^4*x^2),x]

[Out]

(-4*(2 - E))/(E^4*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {(4 (2-e)) \int \frac {1}{x^2} \, dx}{e^4}\\ &=-\frac {4 (2-e)}{e^4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.87 \begin {gather*} -\frac {8-4 e}{e^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 4*E)/(E^4*x^2),x]

[Out]

-((8 - 4*E)/(E^4*x))

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fricas [A]  time = 0.87, size = 11, normalized size = 0.73 \begin {gather*} \frac {4 \, {\left (e - 2\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)+8)/x^2/exp(4),x, algorithm="fricas")

[Out]

4*(e - 2)*e^(-4)/x

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giac [A]  time = 0.34, size = 11, normalized size = 0.73 \begin {gather*} \frac {4 \, {\left (e - 2\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)+8)/x^2/exp(4),x, algorithm="giac")

[Out]

4*(e - 2)*e^(-4)/x

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maple [A]  time = 0.02, size = 14, normalized size = 0.93

method	result	size
gosper	\(\frac {4 \left ({\mathrm e}-2\right ) {\mathrm e}^{-4}}{x}\)	\(14\)
norman	\(\frac {4 \left ({\mathrm e}-2\right ) {\mathrm e}^{-4}}{x}\)	\(14\)
default	\(-\frac {4 \left (2-{\mathrm e}\right ) {\mathrm e}^{-4}}{x}\)	\(16\)
risch	\(\frac {4 \,{\mathrm e}^{-4} {\mathrm e}}{x}-\frac {8 \,{\mathrm e}^{-4}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(1)+8)/x^2/exp(4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.45, size = 11, normalized size = 0.73 \begin {gather*} \frac {4 \, {\left (e - 2\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)+8)/x^2/exp(4),x, algorithm="maxima")

[Out]

4*(e - 2)*e^(-4)/x

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mupad [B]  time = 0.05, size = 11, normalized size = 0.73 \begin {gather*} \frac {4\,{\mathrm {e}}^{-4}\,\left (\mathrm {e}-2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*(4*exp(1) - 8))/x^2,x)

[Out]

(4*exp(-4)*(exp(1) - 2))/x

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sympy [A]  time = 0.06, size = 12, normalized size = 0.80 \begin {gather*} - \frac {8 - 4 e}{x e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(1)+8)/x**2/exp(4),x)

[Out]

-(8 - 4*E)*exp(-4)/x

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