\(\int \frac {7-\log (x)}{x (3+\log (x))} \, dx\) [141]

Optimal result

Integrand size = 16, antiderivative size = 12 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=-\log (x)+10 \log (3+\log (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2412, 45} \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=10 \log (\log (x)+3)-\log (x) \]

[In]

Int[(7 - Log[x])/(x*(3 + Log[x])),x]

[Out]

-Log[x] + 10*Log[3 + Log[x]]

Rule 45

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 2412

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*} \text {integral}& = \text {Subst}\left (\int \frac {7-x}{3+x} \, dx,x,\log (x)\right ) \\ & = \text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=-\log (x)+10 \log (3+\log (x)) \]

[In]

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

[Out]

-Log[x] + 10*Log[3 + Log[x]]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\ln \left (x \right )+10 \ln \left (3+\ln \left (x \right )\right )\)	\(13\)
default	\(-\ln \left (x \right )+10 \ln \left (3+\ln \left (x \right )\right )\)	\(13\)
norman	\(-\ln \left (x \right )+10 \ln \left (3+\ln \left (x \right )\right )\)	\(13\)
risch	\(-\ln \left (x \right )+10 \ln \left (3+\ln \left (x \right )\right )\)	\(13\)
parallelrisch	\(-\ln \left (x \right )+10 \ln \left (3+\ln \left (x \right )\right )\)	\(13\)

[In]

int((7-ln(x))/x/(3+ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=-\log \left (x\right ) + 10 \, \log \left (\log \left (x\right ) + 3\right ) \]

[In]

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

[Out]

-log(x) + 10*log(log(x) + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=- \log {\left (x \right )} + 10 \log {\left (\log {\left (x \right )} + 3 \right )} \]

[In]

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

[Out]

-log(x) + 10*log(log(x) + 3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=-\log \left (x\right ) + 10 \, \log \left (\log \left (x\right ) + 3\right ) \]

[In]

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

[Out]

-log(x) + 10*log(log(x) + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=5 \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) - \log \left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {7-\log (x)}{x (3+\log (x))} \, dx=10\,\ln \left (\ln \left (x\right )+3\right )-\ln \left (x\right ) \]

[In]

int(-(log(x) - 7)/(x*(log(x) + 3)),x)

[Out]

10*log(log(x) + 3) - log(x)