3.69.24 $\int \frac{16+4x-8\log (x)}{(4x+x^2)\log (x)}dx$

Optimal. Leaf size=17 $$2\log \left(\frac{5(4+x){\log }^2(x)}{4x}\right)$$

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Rubi [A]  time = 0.31, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 8, integrand size = 23, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.348, Rules used = {1593, 6741, 12, 6742, 36, 29, 31, 2302} $$\begin{array}{c}-2\log (x)+2\log (x+4)+4\log (\log (x))\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(16 + 4*x - 8*Log[x])/((4*x + x^2)*Log[x]),x]

[Out]

-2*Log[x] + 2*Log[4 + x] + 4*Log[Log[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2302

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{16+4x-8\log (x)}{x(4+x)\log (x)}dx\\ & =\int \frac{4(4+x-2\log (x))}{x(4+x)\log (x)}dx\\ & =4\int \frac{4+x-2\log (x)}{x(4+x)\log (x)}dx\\ & =4\int (-\frac{2}{x(4+x)}+\frac{1}{x\log (x)})dx\\ & =4\int \frac{1}{x\log (x)}dx-8\int \frac{1}{x(4+x)}dx\\ & =-(2\int \frac{1}{x}dx)+2\int \frac{1}{4+x}dx+4\mathrm{Subst}(\int \frac{1}{x}dx,x,\log (x))\\ & =-2\log (x)+2\log (4+x)+4\log (\log (x))\end{array}\end{array}$$

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Mathematica [A]  time = 0.04, size = 20, normalized size = 1.18 $$\begin{array}{c}4(-\frac{\log (x)}{2}+\frac{1}{2}\log (4+x)+\log (\log (x)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(16 + 4*x - 8*Log[x])/((4*x + x^2)*Log[x]),x]

[Out]

4*(-1/2*Log[x] + Log[4 + x]/2 + Log[Log[x]])

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fricas [A]  time = 0.55, size = 16, normalized size = 0.94 $$\begin{array}{c}2\log (x+4)-2\log (x)+4\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)+4*x+16)/(x^2+4*x)/log(x),x, algorithm="fricas")

[Out]

2*log(x + 4) - 2*log(x) + 4*log(log(x))

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giac [A]  time = 0.13, size = 16, normalized size = 0.94 $$\begin{array}{c}2\log (x+4)-2\log (x)+4\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)+4*x+16)/(x^2+4*x)/log(x),x, algorithm="giac")

[Out]

2*log(x + 4) - 2*log(x) + 4*log(log(x))

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maple [A]  time = 0.07, size = 17, normalized size = 1.00

method	result	size
default	$4\ln (\ln (x))-2\ln (x)+2\ln (4+x)$	$17$
norman	$4\ln (\ln (x))-2\ln (x)+2\ln (4+x)$	$17$
risch	$4\ln (\ln (x))-2\ln (x)+2\ln (4+x)$	$17$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*ln(x)+4*x+16)/(x^2+4*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

4*ln(ln(x))-2*ln(x)+2*ln(4+x)

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maxima [A]  time = 0.38, size = 16, normalized size = 0.94 $$\begin{array}{c}2\log (x+4)-2\log (x)+4\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(x)+4*x+16)/(x^2+4*x)/log(x),x, algorithm="maxima")

[Out]

2*log(x + 4) - 2*log(x) + 4*log(log(x))

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mupad [B]  time = 4.12, size = 16, normalized size = 0.94 $$\begin{array}{c}2\ln (x+4)+4\ln (\ln (x))-2\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 8*log(x) + 16)/(log(x)*(4*x + x^2)),x)

[Out]

2*log(x + 4) + 4*log(log(x)) - 2*log(x)

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sympy [A]  time = 0.16, size = 17, normalized size = 1.00 $$\begin{array}{c}-2\log (x)+2\log (x+4)+4\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*ln(x)+4*x+16)/(x**2+4*x)/ln(x),x)

[Out]

-2*log(x) + 2*log(x + 4) + 4*log(log(x))

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