\(\int \log (\sqrt {x}) \, dx\) [139]

Optimal result

Integrand size = 6, antiderivative size = 14 \[ \int \log \left (\sqrt {x}\right ) \, dx=-\frac {x}{2}+x \log \left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2332} \[ \int \log \left (\sqrt {x}\right ) \, dx=x \log \left (\sqrt {x}\right )-\frac {x}{2} \]

[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*} \text {integral}& = -\frac {x}{2}+x \log \left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {1}{2} (-x+x \log (x)) \]

[In]

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

[Out]

(-x + x*Log[x])/2

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71

method	result	size
lookup	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)
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	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)
parallelrisch	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)
parts	\(-\frac {x}{2}+\frac {x \ln \left (x \right )}{2}\)	\(10\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x \log \left (x\right ) - \frac {1}{2} \, x \]

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {x \log {\left (x \right )}}{2} - \frac {x}{2} \]

[In]

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

[Out]

x*log(x)/2 - x/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x \log \left (x\right ) - \frac {1}{2} \, x \]

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {1}{2} \, x \log \left (x\right ) - \frac {1}{2} \, x \]

[In]

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

[Out]

1/2*x*log(x) - 1/2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \log \left (\sqrt {x}\right ) \, dx=\frac {x\,\left (\ln \left (x\right )-1\right )}{2} \]

[In]

int(log(x)/2,x)

[Out]

(x*(log(x) - 1))/2