3.4.21 \(\int \log (\frac {x}{2}) \, dx\) [321]

Optimal. Leaf size=12 \[ -x+x \log \left (\frac {x}{2}\right ) \]

[Out]

-x+x*ln(1/2*x)

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2332} \begin {gather*} x \log \left (\frac {x}{2}\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x/2],x]

[Out]

-x + x*Log[x/2]

Rule 2332

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 \left (\frac {x}{2}\right ) \, dx &=-x+x \log \left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -x+x \log \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[x/2],x]

[Out]

-x + x*Log[x/2]

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Mathics [A]
time = 1.66, size = 8, normalized size = 0.67 \begin {gather*} x \left (-1+\text {Log}\left [\frac {x}{2}\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Log[x/2],x]')

[Out]

x (-1 + Log[x / 2])

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Maple [A]
time = 0.00, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(-x +x \ln \left (\frac {x}{2}\right )\)	\(11\)
default	\(-x +x \ln \left (\frac {x}{2}\right )\)	\(11\)
norman	\(-x +x \ln \left (\frac {x}{2}\right )\)	\(11\)
risch	\(-x +x \ln \left (\frac {x}{2}\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1/2*x),x,method=_RETURNVERBOSE)

[Out]

-x+x*ln(1/2*x)

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Maxima [A]
time = 0.26, size = 10, normalized size = 0.83 \begin {gather*} x \log \left (\frac {1}{2} \, x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/2*x),x, algorithm="maxima")

[Out]

x*log(1/2*x) - x

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Fricas [A]
time = 0.32, size = 10, normalized size = 0.83 \begin {gather*} x \log \left (\frac {1}{2} \, x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/2*x),x, algorithm="fricas")

[Out]

x*log(1/2*x) - x

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Sympy [A]
time = 0.04, size = 7, normalized size = 0.58 \begin {gather*} x \log {\left (\frac {x}{2} \right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1/2*x),x)

[Out]

x*log(x/2) - x

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Giac [A]
time = 0.00, size = 17, normalized size = 1.42 \begin {gather*} 2 \left (\frac {1}{2} x \ln \left (\frac {x}{2}\right )-\frac {x}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1/2*x),x)

[Out]

x*log(1/2*x) - x

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Mupad [B]
time = 0.18, size = 8, normalized size = 0.67 \begin {gather*} x\,\left (\ln \left (\frac {x}{2}\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x/2),x)

[Out]

x*(log(x/2) - 1)

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