3.2.95 \(\int \frac {1+x}{x (x+\log (x))} \, dx\) [195]

Optimal. Leaf size=5 \[ \log (x+\log (x)) \]

[Out]

ln(x+ln(x))

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Rubi [A]
time = 0.03, antiderivative size = 5, normalized size of antiderivative = 1.00, 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 = {6816} \begin {gather*} \log (x+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(x*(x + Log[x])),x]

[Out]

Log[x + Log[x]]

Rule 6816

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 (x+\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \log (x+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(x*(x + Log[x])),x]

[Out]

Log[x + Log[x]]

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Maple [A]
time = 0.10, size = 6, normalized size = 1.20

method	result	size
default	\(\ln \left (x +\ln \left (x \right )\right )\)	\(6\)
norman	\(\ln \left (x +\ln \left (x \right )\right )\)	\(6\)
risch	\(\ln \left (x +\ln \left (x \right )\right )\)	\(6\)
parallelrisch	\(\ln \left (x +\ln \left (x \right )\right )\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+ln(x))

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Maxima [A]
time = 0.38, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x + \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x))

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Fricas [A]
time = 0.57, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x + \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x))

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Sympy [A]
time = 0.04, size = 5, normalized size = 1.00 \begin {gather*} \log {\left (x + \log {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)/x/(x+ln(x)),x)

[Out]

log(x + log(x))

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Giac [A]
time = 0.51, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x + \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x))

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Mupad [B]
time = 0.11, size = 5, normalized size = 1.00 \begin {gather*} \ln \left (x+\ln \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(x*(x + log(x))),x)

[Out]

log(x + log(x))

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((x+1)/x/(x+ln(x)),x)

[Out]

not solved

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