3.50 \(\int \log (\sqrt{x}) \, dx\)

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

[Out]

-x/2 + x*Log[Sqrt[x]]

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Rubi [A]  time = 0.0011754, antiderivative size = 14, 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} \[ x \log \left (\sqrt{x}\right )-\frac{x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0005846, size = 12, normalized size = 0.86 \[ \frac{1}{2} (x \log (x)-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-x + x*Log[x])/2

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Maple [A]  time = 0., size = 10, normalized size = 0.7 \begin{align*} -{\frac{x}{2}}+{\frac{x\ln \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.937011, size = 12, normalized size = 0.86 \begin{align*} \frac{1}{2} \, x \log \left (x\right ) - \frac{1}{2} \, x \end{align*}

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 = 2.215, size = 30, normalized size = 2.14 \begin{align*} \frac{1}{2} \, x \log \left (x\right ) - \frac{1}{2} \, x \end{align*}

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 = 0.079031, size = 8, normalized size = 0.57 \begin{align*} \frac{x \log{\left (x \right )}}{2} - \frac{x}{2} \end{align*}

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 = 1.0501, size = 12, normalized size = 0.86 \begin{align*} \frac{1}{2} \, x \log \left (x\right ) - \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x) - 1/2*x