3.1.16 \(\int \log (x) \, dx\) [16]

Optimal. Leaf size=8 \[ -x+x \log (x) \]

[Out]

-x+x*ln(x)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2332} \begin {gather*} x \log (x)-x \end {gather*}

Antiderivative was successfully verified.

[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*} \int \log (x) \, dx &=-x+x \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -x+x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[x],x]

[Out]

-x + x*Log[x]

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Maple [A]
time = 0.00, size = 9, normalized size = 1.12

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x+x*ln(x)

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Maxima [A]
time = 2.26, size = 8, normalized size = 1.00 \begin {gather*} x \log \left (x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x) - x

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Fricas [A]
time = 0.59, size = 8, normalized size = 1.00 \begin {gather*} x \log \left (x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x) - x

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Sympy [A]
time = 0.02, size = 5, normalized size = 0.62 \begin {gather*} x \log {\left (x \right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x),x)

[Out]

x*log(x) - x

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Giac [A]
time = 0.76, size = 8, normalized size = 1.00 \begin {gather*} x \log \left (x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x),x, algorithm="giac")

[Out]

x*log(x) - x

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Mupad [B]
time = 0.02, size = 6, normalized size = 0.75 \begin {gather*} x\,\left (\ln \left (x\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x),x)

[Out]

x*(log(x) - 1)

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