3.1.50 \(\int \log (\sqrt {x}) \, dx\) [50]

Optimal. Leaf size=14 \[ -\frac {x}{2}+x \log \left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, 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 (\sqrt {x}\right )-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[Sqrt[x]],x]

[Out]

-1/2*x + x*Log[Sqrt[x]]

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{2} (-x+x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[Sqrt[x]],x]

[Out]

(-x + x*Log[x])/2

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Mathics [A]
time = 1.79, size = 7, normalized size = 0.50 \begin {gather*} \frac {x \left (-1+\text {Log}\left [x\right ]\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Log[Sqrt[x]],x]')

[Out]

x (-1 + Log[x]) / 2

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Maple [A]
time = 0.00, size = 10, normalized size = 0.71

method	result	size
default	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)
norman	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)
risch	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 9, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, x \log \left (x\right ) - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

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Fricas [A]
time = 0.32, size = 9, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, x \log \left (x\right ) - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

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Sympy [A]
time = 0.04, size = 8, normalized size = 0.57 \begin {gather*} \frac {x \log {\left (x \right )}}{2} - \frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)/2 - x/2

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Giac [A]
time = 0.00, size = 10, normalized size = 0.71 \begin {gather*} \frac {x \ln x-x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

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Mupad [B]
time = 0.02, size = 7, normalized size = 0.50 \begin {gather*} \frac {x\,\left (\ln \left (x\right )-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/2,x)

[Out]

(x*(log(x) - 1))/2

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