\(\int \frac {8}{(8 x-108 x^2) \log (\frac {2-27 x}{x})+(-2 x+27 x^2) \log (\frac {2-27 x}{x}) \log (\log (\frac {2-27 x}{x}))} \, dx\) [9297]

Optimal result

Integrand size = 56, antiderivative size = 18 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \log \left (4-\log \left (\log \left (5+2 \left (-16+\frac {1}{x}\right )\right )\right )\right ) \]

[Out]

4*ln(4-ln(ln(-27+2/x)))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 6820, 6816} \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \log \left (4-\log \left (\log \left (\frac {2}{x}-27\right )\right )\right ) \]

[In]

Int[8/((8*x - 108*x^2)*Log[(2 - 27*x)/x] + (-2*x + 27*x^2)*Log[(2 - 27*x)/x]*Log[Log[(2 - 27*x)/x]]),x]

[Out]

4*Log[4 - Log[Log[-27 + 2/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 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = 8 \int \frac {1}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx \\ & = 8 \int \frac {1}{(2-27 x) x \log \left (-27+\frac {2}{x}\right ) \left (4-\log \left (\log \left (-27+\frac {2}{x}\right )\right )\right )} \, dx \\ & = 4 \log \left (4-\log \left (\log \left (-27+\frac {2}{x}\right )\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \log \left (-4+\log \left (\log \left (-27+\frac {2}{x}\right )\right )\right ) \]

[In]

Integrate[8/((8*x - 108*x^2)*Log[(2 - 27*x)/x] + (-2*x + 27*x^2)*Log[(2 - 27*x)/x]*Log[Log[(2 - 27*x)/x]]),x]

[Out]

4*Log[-4 + Log[Log[-27 + 2/x]]]

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
default	\(4 \ln \left (\ln \left (\ln \left (\frac {-27 x +2}{x}\right )\right )-4\right )\)	\(17\)
norman	\(4 \ln \left (\ln \left (\ln \left (\frac {-27 x +2}{x}\right )\right )-4\right )\)	\(17\)
parallelrisch	\(4 \ln \left (\ln \left (\ln \left (-\frac {27 x -2}{x}\right )\right )-4\right )\)	\(18\)

[In]

int(8/((27*x^2-2*x)*ln((-27*x+2)/x)*ln(ln((-27*x+2)/x))+(-108*x^2+8*x)*ln((-27*x+2)/x)),x,method=_RETURNVERBOS
E)

[Out]

4*ln(ln(ln((-27*x+2)/x))-4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \, \log \left (\log \left (\log \left (-\frac {27 \, x - 2}{x}\right )\right ) - 4\right ) \]

[In]

integrate(8/((27*x^2-2*x)*log((-27*x+2)/x)*log(log((-27*x+2)/x))+(-108*x^2+8*x)*log((-27*x+2)/x)),x, algorithm
="fricas")

[Out]

4*log(log(log(-(27*x - 2)/x)) - 4)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \log {\left (\log {\left (\log {\left (\frac {2 - 27 x}{x} \right )} \right )} - 4 \right )} \]

[In]

integrate(8/((27*x**2-2*x)*ln((-27*x+2)/x)*ln(ln((-27*x+2)/x))+(-108*x**2+8*x)*ln((-27*x+2)/x)),x)

[Out]

4*log(log(log((2 - 27*x)/x)) - 4)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4 \, \log \left (\log \left (-\log \left (x\right ) + \log \left (-27 \, x + 2\right )\right ) - 4\right ) \]

[In]

integrate(8/((27*x^2-2*x)*log((-27*x+2)/x)*log(log((-27*x+2)/x))+(-108*x^2+8*x)*log((-27*x+2)/x)),x, algorithm
="maxima")

[Out]

4*log(log(-log(x) + log(-27*x + 2)) - 4)

Giac [F]

\[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=\int { \frac {8}{{\left (27 \, x^{2} - 2 \, x\right )} \log \left (-\frac {27 \, x - 2}{x}\right ) \log \left (\log \left (-\frac {27 \, x - 2}{x}\right )\right ) - 4 \, {\left (27 \, x^{2} - 2 \, x\right )} \log \left (-\frac {27 \, x - 2}{x}\right )} \,d x } \]

[In]

integrate(8/((27*x^2-2*x)*log((-27*x+2)/x)*log(log((-27*x+2)/x))+(-108*x^2+8*x)*log((-27*x+2)/x)),x, algorithm
="giac")

[Out]

integrate(8/((27*x^2 - 2*x)*log(-(27*x - 2)/x)*log(log(-(27*x - 2)/x)) - 4*(27*x^2 - 2*x)*log(-(27*x - 2)/x)),
x)

Mupad [B] (verification not implemented)

Time = 14.62 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {8}{\left (8 x-108 x^2\right ) \log \left (\frac {2-27 x}{x}\right )+\left (-2 x+27 x^2\right ) \log \left (\frac {2-27 x}{x}\right ) \log \left (\log \left (\frac {2-27 x}{x}\right )\right )} \, dx=4\,\ln \left (\ln \left (\ln \left (\frac {2}{x}-27\right )\right )-4\right ) \]

[In]

int(8/(log(-(27*x - 2)/x)*(8*x - 108*x^2) - log(log(-(27*x - 2)/x))*log(-(27*x - 2)/x)*(2*x - 27*x^2)),x)

[Out]

4*log(log(log(2/x - 27)) - 4)