3.76 \(\int \frac{\log (e x)}{x} \, dx\)

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

[Out]

Log[e*x]^2/2

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Rubi [A]  time = 0.0061649, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {2301} \[ \frac{1}{2} \log ^2(e x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e*x]^2/2

Rule 2301

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 = 0.0009162, size = 10, normalized size = 1. \[ \frac{1}{2} \log ^2(e x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e*x]^2/2

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Maple [A]  time = 0.058, size = 9, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( ex \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19513, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{2} \, \log \left (e x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81106, size = 22, normalized size = 2.2 \begin{align*} \frac{1}{2} \, \log \left (e x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.086911, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (e x \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*x)**2/2

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Giac [A]  time = 1.21108, size = 12, normalized size = 1.2 \begin{align*} \frac{1}{2} \, \log \left (x e\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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