3.5 \(\int \frac {\log (c x)}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, 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 = {2301} \[ \frac {1}{2} \log ^2(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{2} \log ^2(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*x]^2/2

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fricas [A]  time = 0.41, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \log \left (c x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \log \left (c x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 9, normalized size = 0.90 \[ \frac {\ln \left (c x \right )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.68, size = 8, normalized size = 0.80 \[ \frac {1}{2} \, \log \left (c x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.60, size = 8, normalized size = 0.80 \[ \frac {{\ln \left (c\,x\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x)^2/2

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sympy [A]  time = 0.09, size = 7, normalized size = 0.70 \[ \frac {\log {\left (c x \right )}^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x)**2/2

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