\(\int \frac {\log (e x)}{x} \, dx\) [76]

Optimal result

Integrand size = 8, antiderivative size = 10 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]

[Out]

1/2*ln(e*x)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2338} \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]

[In]

Int[Log[e*x]/x,x]

[Out]

Log[e*x]^2/2

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 {1}{2} \log ^2(e x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]

[In]

Integrate[Log[e*x]/x,x]

[Out]

Log[e*x]^2/2

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\ln \left (e x \right )^{2}}{2}\)	\(9\)
default	\(\frac {\ln \left (e x \right )^{2}}{2}\)	\(9\)
norman	\(\frac {\ln \left (e x \right )^{2}}{2}\)	\(9\)
risch	\(\frac {\ln \left (e x \right )^{2}}{2}\)	\(9\)
parts	\(\ln \left (e x \right ) \ln \left (x \right )-\frac {\ln \left (x \right )^{2}}{2}\)	\(15\)

[In]

int(ln(e*x)/x,x,method=_RETURNVERBOSE)

[Out]

1/2*ln(e*x)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]

[In]

integrate(log(e*x)/x,x, algorithm="fricas")

[Out]

1/2*log(e*x)^2

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\log (e x)}{x} \, dx=\frac {\log {\left (e x \right )}^{2}}{2} \]

[In]

integrate(ln(e*x)/x,x)

[Out]

log(e*x)**2/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]

[In]

integrate(log(e*x)/x,x, algorithm="maxima")

[Out]

1/2*log(e*x)^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]

[In]

integrate(log(e*x)/x,x, algorithm="giac")

[Out]

1/2*log(e*x)^2

Mupad [B] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {{\ln \left (e\,x\right )}^2}{2} \]

[In]

int(log(e*x)/x,x)

[Out]

log(e*x)^2/2