3.98.2 \(\int \frac {16-4 x+4 x \log (\frac {x}{2})}{(-4 x+x^2) \log (\frac {x}{2})} \, dx\)

Optimal. Leaf size=18 \[ 4 \log \left (\frac {484 (4-x)}{\log \left (\frac {x}{2}\right )}\right ) \]

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Rubi [A]  time = 0.39, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1593, 6741, 12, 6742, 2302, 29} \begin {gather*} 4 \log (4-x)-4 \log \left (\log \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*Log[4 - x] - 4*Log[Log[x/2]]

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 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 {gather*} \begin {aligned} \text {integral} &=\int \frac {16-4 x+4 x \log \left (\frac {x}{2}\right )}{(-4+x) x \log \left (\frac {x}{2}\right )} \, dx\\ &=\int \frac {4 \left (-4+x-x \log \left (\frac {x}{2}\right )\right )}{(4-x) x \log \left (\frac {x}{2}\right )} \, dx\\ &=4 \int \frac {-4+x-x \log \left (\frac {x}{2}\right )}{(4-x) x \log \left (\frac {x}{2}\right )} \, dx\\ &=4 \int \left (\frac {1}{-4+x}-\frac {1}{x \log \left (\frac {x}{2}\right )}\right ) \, dx\\ &=4 \log (4-x)-4 \int \frac {1}{x \log \left (\frac {x}{2}\right )} \, dx\\ &=4 \log (4-x)-4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {x}{2}\right )\right )\\ &=4 \log (4-x)-4 \log \left (\log \left (\frac {x}{2}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} 4 \left (\log (4-x)-\log \left (\log \left (\frac {x}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*(Log[4 - x] - Log[Log[x/2]])

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fricas [A]  time = 0.61, size = 14, normalized size = 0.78 \begin {gather*} 4 \, \log \left (x - 4\right ) - 4 \, \log \left (\log \left (\frac {1}{2} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(1/2*x)-4*x+16)/(x^2-4*x)/log(1/2*x),x, algorithm="fricas")

[Out]

4*log(x - 4) - 4*log(log(1/2*x))

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giac [A]  time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} 4 \, \log \left (x - 4\right ) - 4 \, \log \left (\log \left (\frac {1}{2} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(1/2*x)-4*x+16)/(x^2-4*x)/log(1/2*x),x, algorithm="giac")

[Out]

4*log(x - 4) - 4*log(log(1/2*x))

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maple [A]  time = 0.09, size = 15, normalized size = 0.83

method	result	size
norman	\(-4 \ln \left (\ln \left (\frac {x}{2}\right )\right )+4 \ln \left (x -4\right )\)	\(15\)
risch	\(-4 \ln \left (\ln \left (\frac {x}{2}\right )\right )+4 \ln \left (x -4\right )\)	\(15\)
derivativedivides	\(4 \ln \left (\frac {x}{2}-2\right )-4 \ln \left (\ln \left (\frac {x}{2}\right )\right )\)	\(17\)
default	\(4 \ln \left (\frac {x}{2}-2\right )-4 \ln \left (\ln \left (\frac {x}{2}\right )\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*ln(1/2*x)-4*x+16)/(x^2-4*x)/ln(1/2*x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(ln(1/2*x))+4*ln(x-4)

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maxima [A]  time = 0.49, size = 17, normalized size = 0.94 \begin {gather*} 4 \, \log \left (x - 4\right ) - 4 \, \log \left (-\log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*log(1/2*x)-4*x+16)/(x^2-4*x)/log(1/2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.74, size = 14, normalized size = 0.78 \begin {gather*} 4\,\ln \left (x-4\right )-4\,\ln \left (\ln \left (\frac {x}{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*log(x - 4) - 4*log(log(x/2))

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sympy [A]  time = 0.11, size = 14, normalized size = 0.78 \begin {gather*} 4 \log {\left (x - 4 \right )} - 4 \log {\left (\log {\left (\frac {x}{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*ln(1/2*x)-4*x+16)/(x**2-4*x)/ln(1/2*x),x)

[Out]

4*log(x - 4) - 4*log(log(x/2))

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