\(\int -\frac {2 \log (x)}{e^8 x} \, dx\) [2028]

Optimal result

Integrand size = 10, antiderivative size = 19 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-\frac {e^{10}}{9}+\log (3)-\frac {\log ^2(x)}{e^8} \]

[Out]

ln(3)-ln(x)^2/exp(8)-1/9*exp(5)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 2338} \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-\frac {\log ^2(x)}{e^8} \]

[In]

Int[(-2*Log[x])/(E^8*x),x]

[Out]

-(Log[x]^2/E^8)

Rule 12

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

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \int \frac {\log (x)}{x} \, dx}{e^8} \\ & = -\frac {\log ^2(x)}{e^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-\frac {\log ^2(x)}{e^8} \]

[In]

Integrate[(-2*Log[x])/(E^8*x),x]

[Out]

-(Log[x]^2/E^8)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.47

method	result	size
risch	\(-\ln \left (x \right )^{2} {\mathrm e}^{-8}\)	\(9\)
derivativedivides	\(-\ln \left (x \right )^{2} {\mathrm e}^{-8}\)	\(11\)
default	\(-\ln \left (x \right )^{2} {\mathrm e}^{-8}\)	\(11\)
norman	\(-\ln \left (x \right )^{2} {\mathrm e}^{-8}\)	\(11\)
parts	\(-\ln \left (x \right )^{2} {\mathrm e}^{-8}\)	\(11\)

[In]

int(-2*ln(x)/x/exp(8),x,method=_RETURNVERBOSE)

[Out]

-ln(x)^2*exp(-8)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-e^{\left (-8\right )} \log \left (x\right )^{2} \]

[In]

integrate(-2*log(x)/x/exp(8),x, algorithm="fricas")

[Out]

-e^(-8)*log(x)^2

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=- \frac {\log {\left (x \right )}^{2}}{e^{8}} \]

[In]

integrate(-2*ln(x)/x/exp(8),x)

[Out]

-exp(-8)*log(x)**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-e^{\left (-8\right )} \log \left (x\right )^{2} \]

[In]

integrate(-2*log(x)/x/exp(8),x, algorithm="maxima")

[Out]

-e^(-8)*log(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-e^{\left (-8\right )} \log \left (x\right )^{2} \]

[In]

integrate(-2*log(x)/x/exp(8),x, algorithm="giac")

[Out]

-e^(-8)*log(x)^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int -\frac {2 \log (x)}{e^8 x} \, dx=-{\mathrm {e}}^{-8}\,{\ln \left (x\right )}^2 \]

[In]

int(-(2*exp(-8)*log(x))/x,x)

[Out]

-exp(-8)*log(x)^2