### 3.141 $$\int \frac{7-\log (x)}{x (3+\log (x))} \, dx$$

Optimal. Leaf size=12 $10 \log (\log (x)+3)-\log (x)$

[Out]

-Log[x] + 10*Log[3 + Log[x]]

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Rubi [A]  time = 0.0362636, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {2365, 43} $10 \log (\log (x)+3)-\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(7 - Log[x])/(x*(3 + Log[x])),x]

[Out]

-Log[x] + 10*Log[3 + Log[x]]

Rule 2365

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(c_.)*(x_)^(n_.)]*(e_.))^(q_.))/(x_), x_Symbol]
:> Dist[1/n, Subst[Int[(a + b*x)^p*(d + e*x)^q, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{7-\log (x)}{x (3+\log (x))} \, dx &=\operatorname{Subst}\left (\int \frac{7-x}{3+x} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-1+\frac{10}{3+x}\right ) \, dx,x,\log (x)\right )\\ &=-\log (x)+10 \log (3+\log (x))\\ \end{align*}

Mathematica [A]  time = 0.0257176, size = 12, normalized size = 1. $10 \log (\log (x)+3)-\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(7 - Log[x])/(x*(3 + Log[x])),x]

[Out]

-Log[x] + 10*Log[3 + Log[x]]

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Maple [A]  time = 0.006, size = 13, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) +10\,\ln \left ( 3+\ln \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((7-ln(x))/x/(3+ln(x)),x)

[Out]

-ln(x)+10*ln(3+ln(x))

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Maxima [A]  time = 1.09944, size = 16, normalized size = 1.33 \begin{align*} -\log \left (x\right ) + 10 \, \log \left (\log \left (x\right ) + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-log(x))/x/(3+log(x)),x, algorithm="maxima")

[Out]

-log(x) + 10*log(log(x) + 3)

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Fricas [A]  time = 2.03898, size = 41, normalized size = 3.42 \begin{align*} -\log \left (x\right ) + 10 \, \log \left (\log \left (x\right ) + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-log(x))/x/(3+log(x)),x, algorithm="fricas")

[Out]

-log(x) + 10*log(log(x) + 3)

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Sympy [A]  time = 0.107148, size = 10, normalized size = 0.83 \begin{align*} - \log{\left (x \right )} + 10 \log{\left (\log{\left (x \right )} + 3 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-ln(x))/x/(3+ln(x)),x)

[Out]

-log(x) + 10*log(log(x) + 3)

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Giac [B]  time = 1.25913, size = 36, normalized size = 3. \begin{align*} 5 \, \log \left (\frac{1}{4} \, \pi ^{2}{\left (\mathrm{sgn}\left (x\right ) - 1\right )}^{2} +{\left (\log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) - \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((7-log(x))/x/(3+log(x)),x, algorithm="giac")

[Out]

5*log(1/4*pi^2*(sgn(x) - 1)^2 + (log(abs(x)) + 3)^2) - log(x)