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

Optimal. Leaf size=10 \[ \frac {1}{2} \log ^2(e x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2338} \begin {gather*} \frac {1}{2} \log ^2(e x) \end {gather*}

Antiderivative was successfully verified.

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log ^2(e x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e*x]^2/2

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 9, normalized size = 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	\(\frac {\ln \left (e x \right )^{2}}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left (x e\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left (x e\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.02, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (e x \right )}^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*x)**2/2

________________________________________________________________________________________

Giac [A]
time = 4.15, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x)^2 + log(x)

________________________________________________________________________________________

Mupad [B]
time = 0.17, size = 8, normalized size = 0.80 \begin {gather*} \frac {{\ln \left (e\,x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*x)^2/2

________________________________________________________________________________________