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

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

[Out]

-x+x*ln(c*x)

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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 = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ x \log (c x)-x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x],x]

[Out]

-x + x*Log[c*x]

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fricas [A]  time = 0.40, size = 10, normalized size = 1.00 \[ x \log \left (c x\right ) - x \]

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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giac [A]  time = 0.23, size = 16, normalized size = 1.60 \[ \frac {c x \log \left (c x\right ) - c x}{c} \]

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

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maple [A]  time = 0.02, size = 11, normalized size = 1.10 \[ x \ln \left (c x \right )-x \]

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 = 0.56, size = 16, normalized size = 1.60 \[ \frac {c x \log \left (c x\right ) - c x}{c} \]

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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mupad [B]  time = 0.02, size = 8, normalized size = 0.80 \[ x\,\left (\ln \left (c\,x\right )-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x),x)

[Out]

x*(log(c*x) - 1)

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

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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