3.45.74 \(\int -\frac {8}{(1620+144 x+180 \log (2)) \log (\frac {1}{5} (45+4 x+5 \log (2))) \log (\log (\frac {1}{5} (45+4 x+5 \log (2))))+(540+48 x+60 \log (2)) \log (\frac {1}{5} (45+4 x+5 \log (2))) \log (\log (\frac {1}{5} (45+4 x+5 \log (2)))) \log (\log (\log (\frac {1}{5} (45+4 x+5 \log (2)))))+(45+4 x+5 \log (2)) \log (\frac {1}{5} (45+4 x+5 \log (2))) \log (\log (\frac {1}{5} (45+4 x+5 \log (2)))) \log ^2(\log (\log (\frac {1}{5} (45+4 x+5 \log (2)))))} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{3 x+\frac {1}{2} x \log \left (\log \left (\log \left (9+\frac {4 x}{5}+\log (2)\right )\right )\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 18, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 3, integrand size = 156, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {12, 6688, 6686} \begin {gather*} \frac {2}{\log \left (\log \left (\log \left (\frac {4 x}{5}+9+\log (2)\right )\right )\right )+6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-8/((1620 + 144*x + 180*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]] + (540 + 48
*x + 60*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]]*Log[Log[Log[(45 + 4*x + 5*Log[2
])/5]]] + (45 + 4*x + 5*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]]*Log[Log[Log[(45
+ 4*x + 5*Log[2])/5]]]^2),x]

[Out]

2/(6 + Log[Log[Log[9 + (4*x)/5 + Log[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 6686

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]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (8 \int \frac {1}{(1620+144 x+180 \log (2)) \log \left (\frac {1}{5} (45+4 x+5 \log (2))\right ) \log \left (\log \left (\frac {1}{5} (45+4 x+5 \log (2))\right )\right )+(540+48 x+60 \log (2)) \log \left (\frac {1}{5} (45+4 x+5 \log (2))\right ) \log \left (\log \left (\frac {1}{5} (45+4 x+5 \log (2))\right )\right ) \log \left (\log \left (\log \left (\frac {1}{5} (45+4 x+5 \log (2))\right )\right )\right )+(45+4 x+5 \log (2)) \log \left (\frac {1}{5} (45+4 x+5 \log (2))\right ) \log \left (\log \left (\frac {1}{5} (45+4 x+5 \log (2))\right )\right ) \log ^2\left (\log \left (\log \left (\frac {1}{5} (45+4 x+5 \log (2))\right )\right )\right )} \, dx\right )\\ &=-\left (8 \int \frac {1}{(45+4 x+\log (32)) \log \left (9+\frac {4 x}{5}+\log (2)\right ) \log \left (\log \left (9+\frac {4 x}{5}+\log (2)\right )\right ) \left (6+\log \left (\log \left (\log \left (9+\frac {4 x}{5}+\log (2)\right )\right )\right )\right )^2} \, dx\right )\\ &=\frac {2}{6+\log \left (\log \left (\log \left (9+\frac {4 x}{5}+\log (2)\right )\right )\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 18, normalized size = 0.72 \begin {gather*} \frac {2}{6+\log \left (\log \left (\log \left (9+\frac {4 x}{5}+\log (2)\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-8/((1620 + 144*x + 180*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]] + (54
0 + 48*x + 60*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]]*Log[Log[Log[(45 + 4*x + 5
*Log[2])/5]]] + (45 + 4*x + 5*Log[2])*Log[(45 + 4*x + 5*Log[2])/5]*Log[Log[(45 + 4*x + 5*Log[2])/5]]*Log[Log[L
og[(45 + 4*x + 5*Log[2])/5]]]^2),x]

[Out]

2/(6 + Log[Log[Log[9 + (4*x)/5 + Log[2]]]])

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 16, normalized size = 0.64 \begin {gather*} \frac {2}{\log \left (\log \left (\log \left (\frac {4}{5} \, x + \log \relax (2) + 9\right )\right )\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8/((5*log(2)+4*x+45)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))^2+(
60*log(2)+48*x+540)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))+(180*log(2)+144
*x+1620)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))),x, algorithm="fricas")

[Out]

2/(log(log(log(4/5*x + log(2) + 9))) + 6)

________________________________________________________________________________________

giac [A]  time = 0.46, size = 23, normalized size = 0.92 \begin {gather*} \frac {2}{\log \left (\log \left (-\log \relax (5) + \log \left (4 \, x + 5 \, \log \relax (2) + 45\right )\right )\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8/((5*log(2)+4*x+45)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))^2+(
60*log(2)+48*x+540)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))+(180*log(2)+144
*x+1620)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))),x, algorithm="giac")

[Out]

2/(log(log(-log(5) + log(4*x + 5*log(2) + 45))) + 6)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 17, normalized size = 0.68

method	result	size
risch	\(\frac {2}{\ln \left (\ln \left (\ln \left (\ln \relax (2)+\frac {4 x}{5}+9\right )\right )\right )+6}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-8/((5*ln(2)+4*x+45)*ln(ln(2)+4/5*x+9)*ln(ln(ln(2)+4/5*x+9))*ln(ln(ln(ln(2)+4/5*x+9)))^2+(60*ln(2)+48*x+54
0)*ln(ln(2)+4/5*x+9)*ln(ln(ln(2)+4/5*x+9))*ln(ln(ln(ln(2)+4/5*x+9)))+(180*ln(2)+144*x+1620)*ln(ln(2)+4/5*x+9)*
ln(ln(ln(2)+4/5*x+9))),x,method=_RETURNVERBOSE)

[Out]

2/(ln(ln(ln(ln(2)+4/5*x+9)))+6)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} \frac {2}{\log \left (\log \left (-\log \relax (5) + \log \left (4 \, x + 5 \, \log \relax (2) + 45\right )\right )\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8/((5*log(2)+4*x+45)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))^2+(
60*log(2)+48*x+540)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))*log(log(log(log(2)+4/5*x+9)))+(180*log(2)+144
*x+1620)*log(log(2)+4/5*x+9)*log(log(log(2)+4/5*x+9))),x, algorithm="maxima")

[Out]

2/(log(log(-log(5) + log(4*x + 5*log(2) + 45))) + 6)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {8}{\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\,\ln \left (\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\right )\,\left (4\,x+5\,\ln \relax (2)+45\right )\,{\ln \left (\ln \left (\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\right )\right )}^2+\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\,\ln \left (\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\right )\,\left (48\,x+60\,\ln \relax (2)+540\right )\,\ln \left (\ln \left (\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\right )\right )+\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\,\ln \left (\ln \left (\frac {4\,x}{5}+\ln \relax (2)+9\right )\right )\,\left (144\,x+180\,\ln \relax (2)+1620\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-8/(log((4*x)/5 + log(2) + 9)*log(log((4*x)/5 + log(2) + 9))*(144*x + 180*log(2) + 1620) + log((4*x)/5 + l
og(2) + 9)*log(log((4*x)/5 + log(2) + 9))*log(log(log((4*x)/5 + log(2) + 9)))^2*(4*x + 5*log(2) + 45) + log((4
*x)/5 + log(2) + 9)*log(log((4*x)/5 + log(2) + 9))*log(log(log((4*x)/5 + log(2) + 9)))*(48*x + 60*log(2) + 540
)),x)

[Out]

int(-8/(log((4*x)/5 + log(2) + 9)*log(log((4*x)/5 + log(2) + 9))*(144*x + 180*log(2) + 1620) + log((4*x)/5 + l
og(2) + 9)*log(log((4*x)/5 + log(2) + 9))*log(log(log((4*x)/5 + log(2) + 9)))^2*(4*x + 5*log(2) + 45) + log((4
*x)/5 + log(2) + 9)*log(log((4*x)/5 + log(2) + 9))*log(log(log((4*x)/5 + log(2) + 9)))*(48*x + 60*log(2) + 540
)), x)

________________________________________________________________________________________

sympy [A]  time = 0.45, size = 17, normalized size = 0.68 \begin {gather*} \frac {2}{\log {\left (\log {\left (\log {\left (\frac {4 x}{5} + \log {\relax (2 )} + 9 \right )} \right )} \right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8/((5*ln(2)+4*x+45)*ln(ln(2)+4/5*x+9)*ln(ln(ln(2)+4/5*x+9))*ln(ln(ln(ln(2)+4/5*x+9)))**2+(60*ln(2)+
48*x+540)*ln(ln(2)+4/5*x+9)*ln(ln(ln(2)+4/5*x+9))*ln(ln(ln(ln(2)+4/5*x+9)))+(180*ln(2)+144*x+1620)*ln(ln(2)+4/
5*x+9)*ln(ln(ln(2)+4/5*x+9))),x)

[Out]

2/(log(log(log(4*x/5 + log(2) + 9))) + 6)

________________________________________________________________________________________