### 3.321 $$\int \log (\frac{x}{2}) \, dx$$

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

[Out]

-x + x*Log[x/2]

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Rubi [A]  time = 0.0013106, antiderivative size = 12, normalized size of antiderivative = 1., 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 = {2295} $x \log \left (\frac{x}{2}\right )-x$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x/2],x]

[Out]

-x + x*Log[x/2]

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

Mathematica [A]  time = 0.0006773, size = 12, normalized size = 1. $x \log \left (\frac{x}{2}\right )-x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x/2],x]

[Out]

-x + x*Log[x/2]

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} -x+x\ln \left ({\frac{x}{2}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.937229, size = 14, normalized size = 1.17 \begin{align*} x \log \left (\frac{1}{2} \, x\right ) - x \end{align*}

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 = 2.3514, size = 24, normalized size = 2. \begin{align*} x \log \left (\frac{1}{2} \, x\right ) - x \end{align*}

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 = 0.081452, size = 7, normalized size = 0.58 \begin{align*} x \log{\left (\frac{x}{2} \right )} - x \end{align*}

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 = 1.04918, size = 14, normalized size = 1.17 \begin{align*} x \log \left (\frac{1}{2} \, x\right ) - x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(1/2*x) - x