\(\int \frac {18+6 \log (\log (\frac {-4+16 x^3}{9 x^3}))}{(-x+4 x^4) \log (\frac {-4+16 x^3}{9 x^3})} \, dx\) [6840]

Optimal result

Integrand size = 49, antiderivative size = 20 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\left (3+\log \left (\log \left (\frac {4 \left (-1+4 x^3\right )}{9 x^3}\right )\right )\right )^2 \]

[Out]

(3+ln(ln(4/9*(4*x^3-1)/x^3)))^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {1607, 6818} \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\left (\log \left (\log \left (-\frac {4 \left (1-4 x^3\right )}{9 x^3}\right )\right )+3\right )^2 \]

[In]

Int[(18 + 6*Log[Log[(-4 + 16*x^3)/(9*x^3)]])/((-x + 4*x^4)*Log[(-4 + 16*x^3)/(9*x^3)]),x]

[Out]

(3 + Log[Log[(-4*(1 - 4*x^3))/(9*x^3)]])^2

Rule 1607

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 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{x \left (-1+4 x^3\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx \\ & = \left (3+\log \left (\log \left (-\frac {4 \left (1-4 x^3\right )}{9 x^3}\right )\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\left (3+\log \left (\log \left (\frac {4}{9} \left (4-\frac {1}{x^3}\right )\right )\right )\right )^2 \]

[In]

Integrate[(18 + 6*Log[Log[(-4 + 16*x^3)/(9*x^3)]])/((-x + 4*x^4)*Log[(-4 + 16*x^3)/(9*x^3)]),x]

[Out]

(3 + Log[Log[(4*(4 - x^(-3)))/9]])^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(18)=36\).

Time = 0.60 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60

method	result	size
default	\(6 \ln \left (-2 \ln \left (2\right )+2 \ln \left (3\right )-\ln \left (\frac {4 x^{3}-1}{x^{3}}\right )\right )+{\ln \left (2 \ln \left (2\right )-2 \ln \left (3\right )+\ln \left (\frac {4 x^{3}-1}{x^{3}}\right )\right )}^{2}\)	\(52\)

[In]

int((6*ln(ln(1/9*(16*x^3-4)/x^3))+18)/(4*x^4-x)/ln(1/9*(16*x^3-4)/x^3),x,method=_RETURNVERBOSE)

[Out]

6*ln(-2*ln(2)+2*ln(3)-ln((4*x^3-1)/x^3))+ln(2*ln(2)-2*ln(3)+ln((4*x^3-1)/x^3))^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\log \left (\log \left (\frac {4 \, {\left (4 \, x^{3} - 1\right )}}{9 \, x^{3}}\right )\right )^{2} + 6 \, \log \left (\log \left (\frac {4 \, {\left (4 \, x^{3} - 1\right )}}{9 \, x^{3}}\right )\right ) \]

[In]

integrate((6*log(log(1/9*(16*x^3-4)/x^3))+18)/(4*x^4-x)/log(1/9*(16*x^3-4)/x^3),x, algorithm="fricas")

[Out]

log(log(4/9*(4*x^3 - 1)/x^3))^2 + 6*log(log(4/9*(4*x^3 - 1)/x^3))

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\log {\left (\log {\left (\frac {\frac {16 x^{3}}{9} - \frac {4}{9}}{x^{3}} \right )} \right )}^{2} + 6 \log {\left (\log {\left (\frac {\frac {16 x^{3}}{9} - \frac {4}{9}}{x^{3}} \right )} \right )} \]

[In]

integrate((6*ln(ln(1/9*(16*x**3-4)/x**3))+18)/(4*x**4-x)/ln(1/9*(16*x**3-4)/x**3),x)

[Out]

log(log((16*x**3/9 - 4/9)/x**3))**2 + 6*log(log((16*x**3/9 - 4/9)/x**3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.20 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=-\log \left (-2 \, \log \left (3\right ) + 2 \, \log \left (2\right ) + \log \left (4 \, x^{3} - 1\right ) - 3 \, \log \left (x\right )\right )^{2} + 2 \, \log \left (-2 \, \log \left (3\right ) + 2 \, \log \left (2\right ) + \log \left (4 \, x^{3} - 1\right ) - 3 \, \log \left (x\right )\right ) \log \left (\log \left (-\frac {4}{9 \, x^{3}} + \frac {16}{9}\right )\right ) + 6 \, \log \left (-2 \, \log \left (3\right ) + 2 \, \log \left (2\right ) + \log \left (4 \, x^{3} - 1\right ) - 3 \, \log \left (x\right )\right ) \]

[In]

integrate((6*log(log(1/9*(16*x^3-4)/x^3))+18)/(4*x^4-x)/log(1/9*(16*x^3-4)/x^3),x, algorithm="maxima")

[Out]

-log(-2*log(3) + 2*log(2) + log(4*x^3 - 1) - 3*log(x))^2 + 2*log(-2*log(3) + 2*log(2) + log(4*x^3 - 1) - 3*log
(x))*log(log(-4/9/x^3 + 16/9)) + 6*log(-2*log(3) + 2*log(2) + log(4*x^3 - 1) - 3*log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (18) = 36\).

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.85 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=-\log \left (-\log \left (9 \, x^{3}\right ) + \log \left (16 \, x^{3} - 4\right )\right )^{2} + 2 \, \log \left (-\log \left (9 \, x^{3}\right ) + \log \left (16 \, x^{3} - 4\right )\right ) \log \left (\log \left (\frac {4 \, {\left (4 \, x^{3} - 1\right )}}{9 \, x^{3}}\right )\right ) + 6 \, \log \left (-\log \left (9 \, x^{3}\right ) + \log \left (16 \, x^{3} - 4\right )\right ) \]

[In]

integrate((6*log(log(1/9*(16*x^3-4)/x^3))+18)/(4*x^4-x)/log(1/9*(16*x^3-4)/x^3),x, algorithm="giac")

[Out]

-log(-log(9*x^3) + log(16*x^3 - 4))^2 + 2*log(-log(9*x^3) + log(16*x^3 - 4))*log(log(4/9*(4*x^3 - 1)/x^3)) + 6
*log(-log(9*x^3) + log(16*x^3 - 4))

Mupad [B] (verification not implemented)

Time = 13.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {18+6 \log \left (\log \left (\frac {-4+16 x^3}{9 x^3}\right )\right )}{\left (-x+4 x^4\right ) \log \left (\frac {-4+16 x^3}{9 x^3}\right )} \, dx=\ln \left (\ln \left (\frac {\frac {16\,x^3}{9}-\frac {4}{9}}{x^3}\right )\right )\,\left (\ln \left (\ln \left (\frac {\frac {16\,x^3}{9}-\frac {4}{9}}{x^3}\right )\right )+6\right ) \]

[In]

int(-(6*log(log(((16*x^3)/9 - 4/9)/x^3)) + 18)/(log(((16*x^3)/9 - 4/9)/x^3)*(x - 4*x^4)),x)

[Out]

log(log(((16*x^3)/9 - 4/9)/x^3))*(log(log(((16*x^3)/9 - 4/9)/x^3)) + 6)