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

Optimal. Leaf size=9 \[ \log \left (\frac{\log (x)}{x}+1\right ) \]

[Out]

Log[1 + Log[x]/x]

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Rubi [A]  time = 0.0718203, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6712, 31} \[ \log \left (\frac{\log (x)}{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Log[x])/(x*(x + Log[x])),x]

[Out]

Log[1 + Log[x]/x]

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1-\log (x)}{x (x+\log (x))} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\frac{\log (x)}{x}\right )\\ &=\log \left (1+\frac{\log (x)}{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.051494, size = 10, normalized size = 1.11 \[ \log (x+\log (x))-\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[x + Log[x]]

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Maple [A]  time = 0.01, size = 11, normalized size = 1.2 \begin{align*} -\ln \left ( x \right ) +\ln \left ( x+\ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x)+ln(x+ln(x))

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Maxima [A]  time = 1.19336, size = 14, normalized size = 1.56 \begin{align*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x)) - log(x)

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Fricas [A]  time = 2.15737, size = 35, normalized size = 3.89 \begin{align*} \log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x)) - log(x)

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Sympy [A]  time = 0.114662, size = 8, normalized size = 0.89 \begin{align*} - \log{\left (x \right )} + \log{\left (x + \log{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(x + log(x))

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Giac [A]  time = 1.3485, size = 19, normalized size = 2.11 \begin{align*} -\log \left (x\right ) + \log \left (-x - \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(-x - log(x))