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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.03, size = 12, normalized size = 1.00 \[ 10 \log (\log (x)+3)-\log (x) \]

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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fricas [A]  time = 0.86, size = 12, normalized size = 1.00 \[ -\log \relax (x) + 10 \, \log \left (\log \relax (x) + 3\right ) \]

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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giac [B]  time = 0.16, size = 27, normalized size = 2.25 \[ 5 \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) - \log \relax (x) \]

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)

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maple [A]  time = 0.07, size = 13, normalized size = 1.08 \[ -\ln \relax (x )+10 \ln \left (\ln \relax (x )+3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 12, normalized size = 1.00 \[ -\log \relax (x) + 10 \, \log \left (\log \relax (x) + 3\right ) \]

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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mupad [B]  time = 0.37, size = 12, normalized size = 1.00 \[ 10\,\ln \left (\ln \relax (x)+3\right )-\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) - 7)/(x*(log(x) + 3)),x)

[Out]

10*log(log(x) + 3) - log(x)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.83 \[ - \log {\relax (x )} + 10 \log {\left (\log {\relax (x )} + 3 \right )} \]

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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