\(\int \log (x) \, dx\) [144]

Optimal result

Integrand size = 2, antiderivative size = 8 \[ \int \log (x) \, dx=-x+x \log (x) \]

[Out]

x*ln(x)-x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2332} \[ \int \log (x) \, dx=x \log (x)-x \]

[In]

Int[Log[x],x]

[Out]

-x + x*Log[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}& = -x+x \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \log (x) \, dx=-x+x \log (x) \]

[In]

Integrate[Log[x],x]

[Out]

-x + x*Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12

method	result	size
lookup	\(x \ln \left (x \right )-x\)	\(9\)
default	\(x \ln \left (x \right )-x\)	\(9\)
norman	\(x \ln \left (x \right )-x\)	\(9\)
risch	\(x \ln \left (x \right )-x\)	\(9\)
parallelrisch	\(x \ln \left (x \right )-x\)	\(9\)
parts	\(x \ln \left (x \right )-x\)	\(9\)

[In]

int(ln(x),x,method=_RETURNVERBOSE)

[Out]

x*ln(x)-x

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \log (x) \, dx=x \log \left (x\right ) - x \]

[In]

integrate(log(x),x, algorithm="fricas")

[Out]

x*log(x) - x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62 \[ \int \log (x) \, dx=x \log {\left (x \right )} - x \]

[In]

integrate(ln(x),x)

[Out]

x*log(x) - x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \log (x) \, dx=x \log \left (x\right ) - x \]

[In]

integrate(log(x),x, algorithm="maxima")

[Out]

x*log(x) - x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \log (x) \, dx=x \log \left (x\right ) - x \]

[In]

integrate(log(x),x, algorithm="giac")

[Out]

x*log(x) - x

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \log (x) \, dx=x\,\left (\ln \left (x\right )-1\right ) \]

[In]

int(log(x),x)

[Out]

x*(log(x) - 1)