3.102.30 \(\int \frac {1+\log (x)}{-2+x \log (x)} \, dx\)

Optimal. Leaf size=11 \[ \log (2)+\log (2-x \log (x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 8, normalized size of antiderivative = 0.73, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6684} \begin {gather*} \log (2-x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Log[x])/(-2 + x*Log[x]),x]

[Out]

Log[2 - x*Log[x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (2-x \log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 7, normalized size = 0.64 \begin {gather*} \log (-2+x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Log[x])/(-2 + x*Log[x]),x]

[Out]

Log[-2 + x*Log[x]]

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fricas [A]  time = 0.47, size = 14, normalized size = 1.27 \begin {gather*} \log \relax (x) + \log \left (\frac {x \log \relax (x) - 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log((x*log(x) - 2)/x)

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giac [A]  time = 0.23, size = 7, normalized size = 0.64 \begin {gather*} \log \left (x \log \relax (x) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x*log(x) - 2)

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maple [A]  time = 0.02, size = 8, normalized size = 0.73

method	result	size
derivativedivides	\(\ln \left (x \ln \relax (x )-2\right )\)	\(8\)
default	\(\ln \left (x \ln \relax (x )-2\right )\)	\(8\)
norman	\(\ln \left (x \ln \relax (x )-2\right )\)	\(8\)
risch	\(\ln \relax (x )+\ln \left (\ln \relax (x )-\frac {2}{x}\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x*ln(x)-2)

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maxima [A]  time = 0.35, size = 7, normalized size = 0.64 \begin {gather*} \log \left (x \log \relax (x) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x*log(x) - 2)

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mupad [B]  time = 7.52, size = 7, normalized size = 0.64 \begin {gather*} \ln \left (x\,\ln \relax (x)-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) + 1)/(x*log(x) - 2),x)

[Out]

log(x*log(x) - 2)

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sympy [A]  time = 0.12, size = 10, normalized size = 0.91 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\relax (x )} - \frac {2}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(log(x) - 2/x)

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