3.1.56 \(\int \log (\frac {1}{x}) \, dx\) [56]

Optimal. Leaf size=8 \[ x+x \log \left (\frac {1}{x}\right ) \]

[Out]

x+x*ln(1/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 = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2332} \begin {gather*} x+x \log \left (\frac {1}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x^(-1)],x]

[Out]

x + x*Log[x^(-1)]

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 {gather*} \begin {aligned} \text {Integral} &=x+x \log \left (\frac {1}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} x+x \log \left (\frac {1}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[x^(-1)],x]

[Out]

x + x*Log[x^(-1)]

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Maple [A]
time = 0.01, size = 9, normalized size = 1.12

method	result	size
derivativedivides	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)
default	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)
norman	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)
risch	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)
parallelrisch	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)
parts	\(x +x \ln \left (\frac {1}{x}\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+x*ln(1/x)

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Maxima [A]
time = 0.37, size = 7, normalized size = 0.88 \begin {gather*} -x \log \left (x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*log(x) + x

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Fricas [A]
time = 0.58, size = 8, normalized size = 1.00 \begin {gather*} x \log \left (\frac {1}{x}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(1/x) + x

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Sympy [A]
time = 0.03, size = 7, normalized size = 0.88 \begin {gather*} x \log {\left (\frac {1}{x} \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1/x),x)

[Out]

x*log(1/x) + x

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Giac [A]
time = 0.49, size = 7, normalized size = 0.88 \begin {gather*} -x \log \left (x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*log(x) + x

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Mupad [B]
time = 0.02, size = 8, normalized size = 1.00 \begin {gather*} x\,\left (\ln \left (\frac {1}{x}\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(1/x),x)

[Out]

x*(log(1/x) + 1)

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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 1.12 \begin {gather*} x \ln \left (\frac {1}{x}\right )+\ln \left (x \right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(ln(1/x),x)

[Out]

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

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