3.3.18 \(\int \frac {e^{-x} (-1+(1-x) \log (x))}{\log ^2(x)} \, dx\) [218]

Optimal. Leaf size=11 \[ \frac {e^{-x} x}{\log (x)} \]

[Out]

x/exp(x)/ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2232} \begin {gather*} \frac {e^{-x} x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + (1 - x)*Log[x])/(E^x*Log[x]^2),x]

[Out]

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

Rule 2232

Int[Log[(d_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((e_) + Log[(d_.)*(x_)]*(h_.)*((f_.) + (g_.)*(x_))
), x_Symbol] :> Simp[e*x*F^(c*(a + b*x))*(Log[d*x]^(n + 1)/(n + 1)), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, n
}, x] && EqQ[e - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {e^{-x} (-1+(1-x) \log (x))}{\log ^2(x)} \, dx &=\frac {e^{-x} x}{\log (x)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {e^{-x} x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + (1 - x)*Log[x])/(E^x*Log[x]^2),x]

[Out]

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

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Mathics [A]
time = 1.74, size = 11, normalized size = 1.00 \begin {gather*} \frac {x E^{-x}}{\text {Log}\left [x\right ]} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(Log[x]*(1-x)-1)/(E^x*Log[x]^2),x]')

[Out]

x E ^ (-x) / Log[x]

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Maple [A]
time = 0.01, size = 11, normalized size = 1.00

method	result	size
norman	\(\frac {x \,{\mathrm e}^{-x}}{\ln \left (x \right )}\)	\(11\)
risch	\(\frac {x \,{\mathrm e}^{-x}}{\ln \left (x \right )}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/exp(x)/ln(x)

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.91 \begin {gather*} \frac {x e^{\left (-x\right )}}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.31, size = 10, normalized size = 0.91 \begin {gather*} \frac {x e^{\left (-x\right )}}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 7, normalized size = 0.64 \begin {gather*} \frac {x e^{- x}}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 8, normalized size = 0.73 \begin {gather*} \frac {x \mathrm {e}^{-x}}{\ln x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+(1-x)*log(x))/exp(x)/log(x)^2,x)

[Out]

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

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Mupad [B]
time = 0.22, size = 10, normalized size = 0.91 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(log(x)*(x - 1) + 1))/log(x)^2,x)

[Out]

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

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