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

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

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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} $$\begin{array}{c}\frac{3e^{-x}\log (x)}{x^3}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{3e^{-x}(1-3\log (x)-x\log (x))}{x^4}dx\\ & =3\int \frac{e^{-x}(1-3\log (x)-x\log (x))}{x^4}dx\\ & =\frac{3e^{-x}\log (x)}{x^3}\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{3e^{(-x)}\log (x)}{x^3}\end{array}$$

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 $$\begin{array}{c}\frac{3e^{(-x)}\log (x)}{x^3}\end{array}$$

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	$\frac{3\ln (x){\mathrm{e}}^{-x}}{x^3}$	$12$
risch	$\frac{3\ln (x){\mathrm{e}}^{-x}}{x^3}$	$12$

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 $$\begin{array}{c}\frac{3e^{(-x)}\log (x)}{x^3}-3\mathrm{\Gamma }(-3,x)-3\int \frac{e^{(-x)}}{x^4}dx\end{array}$$

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 $$\begin{array}{c}\frac{3{\mathrm{e}}^{-x}\ln (x)}{x^3}\end{array}$$

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 $$\begin{array}{c}\frac{3e^{-x}\log (x)}{x^3}\end{array}$$

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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