3.4 \(\int \log (c x) \, dx\)

Optimal. Leaf size=10 \[ x \log (c x)-x \]

[Out]

-x + x*Log[c*x]

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Rubi [A]  time = 0.0013316, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2295} \[ x \log (c x)-x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x],x]

[Out]

-x + x*Log[c*x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \log (c x) \, dx &=-x+x \log (c x)\\ \end{align*}

Mathematica [A]  time = 0.0006168, size = 10, normalized size = 1. \[ x \log (c x)-x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x],x]

[Out]

-x + x*Log[c*x]

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Maple [A]  time = 0.036, size = 11, normalized size = 1.1 \begin{align*} -x+x\ln \left ( cx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x),x)

[Out]

-x+x*ln(c*x)

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Maxima [A]  time = 0.989558, size = 22, normalized size = 2.2 \begin{align*} \frac{c x \log \left (c x\right ) - c x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x*log(c*x) - c*x)/c

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Fricas [A]  time = 0.845204, size = 22, normalized size = 2.2 \begin{align*} x \log \left (c x\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x) - x

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Sympy [A]  time = 0.089146, size = 7, normalized size = 0.7 \begin{align*} x \log{\left (c x \right )} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x),x)

[Out]

x*log(c*x) - x

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Giac [A]  time = 1.07016, size = 22, normalized size = 2.2 \begin{align*} \frac{c x \log \left (c x\right ) - c x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x*log(c*x) - c*x)/c