3.321 \(\int \log \left (\frac{x}{2}\right ) \, dx\)

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 = 0.00460648, 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 \[ x \log \left (\frac{x}{2}\right )-x \]

Antiderivative was successfully verified.

[In]  Int[Log[x/2],x]

[Out]

-x + x*Log[x/2]

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Rubi in Sympy [A]  time = 0.464809, size = 7, normalized size = 0.58 \[ x \log{\left (\frac{x}{2} \right )} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(1/2*x),x)

[Out]

x*log(x/2) - x

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Mathematica [A]  time = 0.00153016, 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 = 0.003, size = 11, normalized size = 0.9 \[ -x+x\ln \left ({\frac{x}{2}} \right ) \]

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 = 1.32875, size = 14, normalized size = 1.17 \[ x \log \left (\frac{1}{2} \, x\right ) - x \]

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 = 0.23654, size = 14, normalized size = 1.17 \[ x \log \left (\frac{1}{2} \, x\right ) - x \]

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 = 0.069657, size = 7, normalized size = 0.58 \[ x \log{\left (\frac{x}{2} \right )} - x \]

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/XCAS [A]  time = 0.21301, size = 14, normalized size = 1.17 \[ x{\rm ln}\left (\frac{1}{2} \, x\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*ln(1/2*x) - x