3.33.68 \(\int \frac {e^{-x} (3+(-9-3 x) \log (x))}{x^4} \, dx\)

Optimal. Leaf size=14 \[ 5+\frac {3 e^{-x} \log (x)}{x^3} \]

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Rubi [A]  time = 0.18, antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6741, 12, 2288} \begin {gather*} \frac {3 e^{-x} \log (x)}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + (-9 - 3*x)*Log[x])/(E^x*x^4),x]

[Out]

(3*Log[x])/(E^x*x^3)

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{-x} (1-3 \log (x)-x \log (x))}{x^4} \, dx\\ &=3 \int \frac {e^{-x} (1-3 \log (x)-x \log (x))}{x^4} \, dx\\ &=\frac {3 e^{-x} \log (x)}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 12, normalized size = 0.86 \begin {gather*} \frac {3 e^{-x} \log (x)}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + (-9 - 3*x)*Log[x])/(E^x*x^4),x]

[Out]

(3*Log[x])/(E^x*x^3)

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fricas [A]  time = 0.69, size = 11, normalized size = 0.79 \begin {gather*} \frac {3 \, e^{\left (-x\right )} \log \relax (x)}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-9)*log(x)+3)/exp(x)/x^4,x, algorithm="fricas")

[Out]

3*e^(-x)*log(x)/x^3

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giac [A]  time = 0.21, size = 11, normalized size = 0.79 \begin {gather*} \frac {3 \, e^{\left (-x\right )} \log \relax (x)}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-9)*log(x)+3)/exp(x)/x^4,x, algorithm="giac")

[Out]

3*e^(-x)*log(x)/x^3

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maple [A]  time = 0.06, size = 12, normalized size = 0.86

method	result	size
norman	\(\frac {3 \ln \relax (x ) {\mathrm e}^{-x}}{x^{3}}\)	\(12\)
risch	\(\frac {3 \ln \relax (x ) {\mathrm e}^{-x}}{x^{3}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x-9)*ln(x)+3)/exp(x)/x^4,x,method=_RETURNVERBOSE)

[Out]

3*ln(x)/x^3/exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3 \, e^{\left (-x\right )} \log \relax (x)}{x^{3}} - 3 \, \Gamma \left (-3, x\right ) - 3 \, \int \frac {e^{\left (-x\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-9)*log(x)+3)/exp(x)/x^4,x, algorithm="maxima")

[Out]

3*e^(-x)*log(x)/x^3 - 3*gamma(-3, x) - 3*integrate(e^(-x)/x^4, x)

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mupad [B]  time = 1.96, size = 11, normalized size = 0.79 \begin {gather*} \frac {3\,{\mathrm {e}}^{-x}\,\ln \relax (x)}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(log(x)*(3*x + 9) - 3))/x^4,x)

[Out]

(3*exp(-x)*log(x))/x^3

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sympy [A]  time = 0.27, size = 10, normalized size = 0.71 \begin {gather*} \frac {3 e^{- x} \log {\relax (x )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-9)*ln(x)+3)/exp(x)/x**4,x)

[Out]

3*exp(-x)*log(x)/x**3

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