\(\int \frac {432-2160 x+3600 x^2-2000 x^3+(-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5) \log (5 x)+(-1728+5760 x-4800 x^2+(432-6480 x+21600 x^2-26000 x^3+10000 x^4) \log (5 x)) \log (\log (5 x))+(2304-3840 x+(-864+10080 x-21600 x^2+12000 x^3) \log (5 x)) \log ^2(\log (5 x))+(-1024+(768-6400 x+6400 x^2) \log (5 x)) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{(405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5) \log (5 x)+(-2160 x+10800 x^2-18000 x^3+10000 x^4) \log (5 x) \log (\log (5 x))+(4320 x-14400 x^2+12000 x^3) \log (5 x) \log ^2(\log (5 x))+(-3840 x+6400 x^2) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx\) [8715]

Optimal result

Integrand size = 284, antiderivative size = 30 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x-\frac {1}{5} \log \left (\frac {1}{4} x \left (x-(-3+x+4 (x+\log (\log (5 x))))^4\right )\right ) \]

[Out]

x-1/5*ln(1/4*(x-(4*ln(ln(5*x))+5*x-3)^4)*x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(30)=60\).

Time = 8.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.23, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6873, 12, 6820, 6874, 45, 6816} \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=-\frac {1}{5} \log \left (625 x^4-1500 x^3+2000 x^3 \log (\log (5 x))+1350 x^2+2400 x^2 \log ^2(\log (5 x))-3600 x^2 \log (\log (5 x))-541 x+256 \log ^4(\log (5 x))+1280 x \log ^3(\log (5 x))-768 \log ^3(\log (5 x))-2880 x \log ^2(\log (5 x))+864 \log ^2(\log (5 x))+2160 x \log (\log (5 x))-432 \log (\log (5 x))+81\right )+x-\frac {\log (x)}{5} \]

[In]

Int[(432 - 2160*x + 3600*x^2 - 2000*x^3 + (-81 + 1487*x - 6755*x^2 + 12750*x^3 - 10625*x^4 + 3125*x^5)*Log[5*x
] + (-1728 + 5760*x - 4800*x^2 + (432 - 6480*x + 21600*x^2 - 26000*x^3 + 10000*x^4)*Log[5*x])*Log[Log[5*x]] +
(2304 - 3840*x + (-864 + 10080*x - 21600*x^2 + 12000*x^3)*Log[5*x])*Log[Log[5*x]]^2 + (-1024 + (768 - 6400*x +
6400*x^2)*Log[5*x])*Log[Log[5*x]]^3 + (-256 + 1280*x)*Log[5*x]*Log[Log[5*x]]^4)/((405*x - 2705*x^2 + 6750*x^3
- 7500*x^4 + 3125*x^5)*Log[5*x] + (-2160*x + 10800*x^2 - 18000*x^3 + 10000*x^4)*Log[5*x]*Log[Log[5*x]] + (432
0*x - 14400*x^2 + 12000*x^3)*Log[5*x]*Log[Log[5*x]]^2 + (-3840*x + 6400*x^2)*Log[5*x]*Log[Log[5*x]]^3 + 1280*x
*Log[5*x]*Log[Log[5*x]]^4),x]

[Out]

x - Log[x]/5 - Log[81 - 541*x + 1350*x^2 - 1500*x^3 + 625*x^4 - 432*Log[Log[5*x]] + 2160*x*Log[Log[5*x]] - 360
0*x^2*Log[Log[5*x]] + 2000*x^3*Log[Log[5*x]] + 864*Log[Log[5*x]]^2 - 2880*x*Log[Log[5*x]]^2 + 2400*x^2*Log[Log
[5*x]]^2 - 768*Log[Log[5*x]]^3 + 1280*x*Log[Log[5*x]]^3 + 256*Log[Log[5*x]]^4]/5

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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]]

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{5 x \log (5 x) \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )} \, dx \\ & = \frac {1}{5} \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{x \log (5 x) \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )} \, dx \\ & = \frac {1}{5} \int \frac {-16 (-3+5 x+4 \log (\log (5 x)))^3+\log (5 x) \left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5+16 (3-5 x)^2 \left (3-35 x+25 x^2\right ) \log (\log (5 x))+96 \left (-9+105 x-225 x^2+125 x^3\right ) \log ^2(\log (5 x))+256 \left (3-25 x+25 x^2\right ) \log ^3(\log (5 x))+256 (-1+5 x) \log ^4(\log (5 x))\right )}{x \log (5 x) \left (81-541 x+1350 x^2-1500 x^3+625 x^4+16 (-3+5 x)^3 \log (\log (5 x))+96 (3-5 x)^2 \log ^2(\log (5 x))+256 (-3+5 x) \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )} \, dx \\ & = \frac {1}{5} \int \left (\frac {-1+5 x}{x}+\frac {432-2160 x+3600 x^2-2000 x^3+541 x \log (5 x)-2700 x^2 \log (5 x)+4500 x^3 \log (5 x)-2500 x^4 \log (5 x)-1728 \log (\log (5 x))+5760 x \log (\log (5 x))-4800 x^2 \log (\log (5 x))-2160 x \log (5 x) \log (\log (5 x))+7200 x^2 \log (5 x) \log (\log (5 x))-6000 x^3 \log (5 x) \log (\log (5 x))+2304 \log ^2(\log (5 x))-3840 x \log ^2(\log (5 x))+2880 x \log (5 x) \log ^2(\log (5 x))-4800 x^2 \log (5 x) \log ^2(\log (5 x))-1024 \log ^3(\log (5 x))-1280 x \log (5 x) \log ^3(\log (5 x))}{x \log (5 x) \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )}\right ) \, dx \\ & = \frac {1}{5} \int \frac {-1+5 x}{x} \, dx+\frac {1}{5} \int \frac {432-2160 x+3600 x^2-2000 x^3+541 x \log (5 x)-2700 x^2 \log (5 x)+4500 x^3 \log (5 x)-2500 x^4 \log (5 x)-1728 \log (\log (5 x))+5760 x \log (\log (5 x))-4800 x^2 \log (\log (5 x))-2160 x \log (5 x) \log (\log (5 x))+7200 x^2 \log (5 x) \log (\log (5 x))-6000 x^3 \log (5 x) \log (\log (5 x))+2304 \log ^2(\log (5 x))-3840 x \log ^2(\log (5 x))+2880 x \log (5 x) \log ^2(\log (5 x))-4800 x^2 \log (5 x) \log ^2(\log (5 x))-1024 \log ^3(\log (5 x))-1280 x \log (5 x) \log ^3(\log (5 x))}{x \log (5 x) \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )} \, dx \\ & = -\frac {1}{5} \log \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right )+\frac {1}{5} \int \left (5-\frac {1}{x}\right ) \, dx \\ & = x-\frac {\log (x)}{5}-\frac {1}{5} \log \left (81-541 x+1350 x^2-1500 x^3+625 x^4-432 \log (\log (5 x))+2160 x \log (\log (5 x))-3600 x^2 \log (\log (5 x))+2000 x^3 \log (\log (5 x))+864 \log ^2(\log (5 x))-2880 x \log ^2(\log (5 x))+2400 x^2 \log ^2(\log (5 x))-768 \log ^3(\log (5 x))+1280 x \log ^3(\log (5 x))+256 \log ^4(\log (5 x))\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(30)=60\).

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.37 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=\frac {1}{5} \left (5 x-\log (5 x)-\log \left (405-2705 x+6750 x^2-7500 x^3+3125 x^4-2160 \log (\log (5 x))+10800 x \log (\log (5 x))-18000 x^2 \log (\log (5 x))+10000 x^3 \log (\log (5 x))+4320 \log ^2(\log (5 x))-14400 x \log ^2(\log (5 x))+12000 x^2 \log ^2(\log (5 x))-3840 \log ^3(\log (5 x))+6400 x \log ^3(\log (5 x))+1280 \log ^4(\log (5 x))\right )\right ) \]

[In]

Integrate[(432 - 2160*x + 3600*x^2 - 2000*x^3 + (-81 + 1487*x - 6755*x^2 + 12750*x^3 - 10625*x^4 + 3125*x^5)*L
og[5*x] + (-1728 + 5760*x - 4800*x^2 + (432 - 6480*x + 21600*x^2 - 26000*x^3 + 10000*x^4)*Log[5*x])*Log[Log[5*
x]] + (2304 - 3840*x + (-864 + 10080*x - 21600*x^2 + 12000*x^3)*Log[5*x])*Log[Log[5*x]]^2 + (-1024 + (768 - 64
00*x + 6400*x^2)*Log[5*x])*Log[Log[5*x]]^3 + (-256 + 1280*x)*Log[5*x]*Log[Log[5*x]]^4)/((405*x - 2705*x^2 + 67
50*x^3 - 7500*x^4 + 3125*x^5)*Log[5*x] + (-2160*x + 10800*x^2 - 18000*x^3 + 10000*x^4)*Log[5*x]*Log[Log[5*x]]
+ (4320*x - 14400*x^2 + 12000*x^3)*Log[5*x]*Log[Log[5*x]]^2 + (-3840*x + 6400*x^2)*Log[5*x]*Log[Log[5*x]]^3 +
1280*x*Log[5*x]*Log[Log[5*x]]^4),x]

[Out]

(5*x - Log[5*x] - Log[405 - 2705*x + 6750*x^2 - 7500*x^3 + 3125*x^4 - 2160*Log[Log[5*x]] + 10800*x*Log[Log[5*x
]] - 18000*x^2*Log[Log[5*x]] + 10000*x^3*Log[Log[5*x]] + 4320*Log[Log[5*x]]^2 - 14400*x*Log[Log[5*x]]^2 + 1200
0*x^2*Log[Log[5*x]]^2 - 3840*Log[Log[5*x]]^3 + 6400*x*Log[Log[5*x]]^3 + 1280*Log[Log[5*x]]^4])/5

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(26)=52\).

Time = 1.86 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.97

method	result	size
risch	\(x -\frac {\ln \left (x \right )}{5}-\frac {\ln \left (\ln \left (\ln \left (5 x \right )\right )^{4}+\left (5 x -3\right ) \ln \left (\ln \left (5 x \right )\right )^{3}+\left (\frac {75}{8} x^{2}-\frac {45}{4} x +\frac {27}{8}\right ) \ln \left (\ln \left (5 x \right )\right )^{2}+\left (\frac {125}{16} x^{3}-\frac {225}{16} x^{2}+\frac {135}{16} x -\frac {27}{16}\right ) \ln \left (\ln \left (5 x \right )\right )+\frac {625 x^{4}}{256}-\frac {375 x^{3}}{64}+\frac {675 x^{2}}{128}-\frac {541 x}{256}+\frac {81}{256}\right )}{5}\)	\(89\)
default	\(-\frac {\ln \left (x \right )}{5}+x -\frac {\ln \left (\ln \left (\ln \left (5\right )+\ln \left (x \right )\right )^{4}+\left (5 x -3\right ) \ln \left (\ln \left (5\right )+\ln \left (x \right )\right )^{3}+\left (\frac {75}{8} x^{2}-\frac {45}{4} x +\frac {27}{8}\right ) \ln \left (\ln \left (5\right )+\ln \left (x \right )\right )^{2}+\left (\frac {125}{16} x^{3}-\frac {225}{16} x^{2}+\frac {135}{16} x -\frac {27}{16}\right ) \ln \left (\ln \left (5\right )+\ln \left (x \right )\right )+\frac {625 x^{4}}{256}-\frac {375 x^{3}}{64}+\frac {675 x^{2}}{128}-\frac {541 x}{256}+\frac {81}{256}\right )}{5}\)	\(93\)
parallelrisch	\(x -\frac {\ln \left (5 x \right )}{5}-\frac {\ln \left (x^{4}+\frac {16 \ln \left (\ln \left (5 x \right )\right ) x^{3}}{5}+\frac {96 \ln \left (\ln \left (5 x \right )\right )^{2} x^{2}}{25}+\frac {256 \ln \left (\ln \left (5 x \right )\right )^{3} x}{125}+\frac {256 \ln \left (\ln \left (5 x \right )\right )^{4}}{625}-\frac {12 x^{3}}{5}-\frac {144 \ln \left (\ln \left (5 x \right )\right ) x^{2}}{25}-\frac {576 \ln \left (\ln \left (5 x \right )\right )^{2} x}{125}-\frac {768 \ln \left (\ln \left (5 x \right )\right )^{3}}{625}+\frac {54 x^{2}}{25}+\frac {432 \ln \left (\ln \left (5 x \right )\right ) x}{125}+\frac {864 \ln \left (\ln \left (5 x \right )\right )^{2}}{625}-\frac {541 x}{625}-\frac {432 \ln \left (\ln \left (5 x \right )\right )}{625}+\frac {81}{625}\right )}{5}\)	\(124\)

[In]

int(((1280*x-256)*ln(5*x)*ln(ln(5*x))^4+((6400*x^2-6400*x+768)*ln(5*x)-1024)*ln(ln(5*x))^3+((12000*x^3-21600*x
^2+10080*x-864)*ln(5*x)-3840*x+2304)*ln(ln(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*ln(5*x)-4800*x^
2+5760*x-1728)*ln(ln(5*x))+(3125*x^5-10625*x^4+12750*x^3-6755*x^2+1487*x-81)*ln(5*x)-2000*x^3+3600*x^2-2160*x+
432)/(1280*x*ln(5*x)*ln(ln(5*x))^4+(6400*x^2-3840*x)*ln(5*x)*ln(ln(5*x))^3+(12000*x^3-14400*x^2+4320*x)*ln(5*x
)*ln(ln(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-2160*x)*ln(5*x)*ln(ln(5*x))+(3125*x^5-7500*x^4+6750*x^3-2705*x^
2+405*x)*ln(5*x)),x,method=_RETURNVERBOSE)

[Out]

x-1/5*ln(x)-1/5*ln(ln(ln(5*x))^4+(5*x-3)*ln(ln(5*x))^3+(75/8*x^2-45/4*x+27/8)*ln(ln(5*x))^2+(125/16*x^3-225/16
*x^2+135/16*x-27/16)*ln(ln(5*x))+625/256*x^4-375/64*x^3+675/128*x^2-541/256*x+81/256)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (26) = 52\).

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (625 \, x^{4} + 256 \, {\left (5 \, x - 3\right )} \log \left (\log \left (5 \, x\right )\right )^{3} + 256 \, \log \left (\log \left (5 \, x\right )\right )^{4} - 1500 \, x^{3} + 96 \, {\left (25 \, x^{2} - 30 \, x + 9\right )} \log \left (\log \left (5 \, x\right )\right )^{2} + 1350 \, x^{2} + 16 \, {\left (125 \, x^{3} - 225 \, x^{2} + 135 \, x - 27\right )} \log \left (\log \left (5 \, x\right )\right ) - 541 \, x + 81\right ) - \frac {1}{5} \, \log \left (5 \, x\right ) \]

[In]

integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*log(5*x)-1024)*log(log(5*x))^3+((12000
*x^3-21600*x^2+10080*x-864)*log(5*x)-3840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*
log(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6755*x^2+1487*x-81)*log(5*x)-2000*x
^3+3600*x^2-2160*x+432)/(1280*x*log(5*x)*log(log(5*x))^4+(6400*x^2-3840*x)*log(5*x)*log(log(5*x))^3+(12000*x^3
-14400*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-2160*x)*log(5*x)*log(log(5*x))+(312
5*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*log(5*x)),x, algorithm="fricas")

[Out]

x - 1/5*log(625*x^4 + 256*(5*x - 3)*log(log(5*x))^3 + 256*log(log(5*x))^4 - 1500*x^3 + 96*(25*x^2 - 30*x + 9)*
log(log(5*x))^2 + 1350*x^2 + 16*(125*x^3 - 225*x^2 + 135*x - 27)*log(log(5*x)) - 541*x + 81) - 1/5*log(5*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).

Time = 0.41 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {\log {\left (x \right )}}{5} - \frac {\log {\left (\frac {625 x^{4}}{256} - \frac {375 x^{3}}{64} + \frac {675 x^{2}}{128} - \frac {541 x}{256} + \left (5 x - 3\right ) \log {\left (\log {\left (5 x \right )} \right )}^{3} + \left (\frac {75 x^{2}}{8} - \frac {45 x}{4} + \frac {27}{8}\right ) \log {\left (\log {\left (5 x \right )} \right )}^{2} + \left (\frac {125 x^{3}}{16} - \frac {225 x^{2}}{16} + \frac {135 x}{16} - \frac {27}{16}\right ) \log {\left (\log {\left (5 x \right )} \right )} + \log {\left (\log {\left (5 x \right )} \right )}^{4} + \frac {81}{256} \right )}}{5} \]

[In]

integrate(((1280*x-256)*ln(5*x)*ln(ln(5*x))**4+((6400*x**2-6400*x+768)*ln(5*x)-1024)*ln(ln(5*x))**3+((12000*x*
*3-21600*x**2+10080*x-864)*ln(5*x)-3840*x+2304)*ln(ln(5*x))**2+((10000*x**4-26000*x**3+21600*x**2-6480*x+432)*
ln(5*x)-4800*x**2+5760*x-1728)*ln(ln(5*x))+(3125*x**5-10625*x**4+12750*x**3-6755*x**2+1487*x-81)*ln(5*x)-2000*
x**3+3600*x**2-2160*x+432)/(1280*x*ln(5*x)*ln(ln(5*x))**4+(6400*x**2-3840*x)*ln(5*x)*ln(ln(5*x))**3+(12000*x**
3-14400*x**2+4320*x)*ln(5*x)*ln(ln(5*x))**2+(10000*x**4-18000*x**3+10800*x**2-2160*x)*ln(5*x)*ln(ln(5*x))+(312
5*x**5-7500*x**4+6750*x**3-2705*x**2+405*x)*ln(5*x)),x)

[Out]

x - log(x)/5 - log(625*x**4/256 - 375*x**3/64 + 675*x**2/128 - 541*x/256 + (5*x - 3)*log(log(5*x))**3 + (75*x*
*2/8 - 45*x/4 + 27/8)*log(log(5*x))**2 + (125*x**3/16 - 225*x**2/16 + 135*x/16 - 27/16)*log(log(5*x)) + log(lo
g(5*x))**4 + 81/256)/5

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (26) = 52\).

Time = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.13 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (\frac {625}{256} \, x^{4} + {\left (5 \, x - 3\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{3} + \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{4} - \frac {375}{64} \, x^{3} + \frac {3}{8} \, {\left (25 \, x^{2} - 30 \, x + 9\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{2} + \frac {675}{128} \, x^{2} + \frac {1}{16} \, {\left (125 \, x^{3} - 225 \, x^{2} + 135 \, x - 27\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right ) - \frac {541}{256} \, x + \frac {81}{256}\right ) - \frac {1}{5} \, \log \left (x\right ) \]

[In]

integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*log(5*x)-1024)*log(log(5*x))^3+((12000
*x^3-21600*x^2+10080*x-864)*log(5*x)-3840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*
log(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6755*x^2+1487*x-81)*log(5*x)-2000*x
^3+3600*x^2-2160*x+432)/(1280*x*log(5*x)*log(log(5*x))^4+(6400*x^2-3840*x)*log(5*x)*log(log(5*x))^3+(12000*x^3
-14400*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-2160*x)*log(5*x)*log(log(5*x))+(312
5*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*log(5*x)),x, algorithm="maxima")

[Out]

x - 1/5*log(625/256*x^4 + (5*x - 3)*log(log(5) + log(x))^3 + log(log(5) + log(x))^4 - 375/64*x^3 + 3/8*(25*x^2
- 30*x + 9)*log(log(5) + log(x))^2 + 675/128*x^2 + 1/16*(125*x^3 - 225*x^2 + 135*x - 27)*log(log(5) + log(x))
- 541/256*x + 81/256) - 1/5*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (26) = 52\).

Time = 0.52 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.10 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (625 \, x^{4} + 2000 \, x^{3} \log \left (\log \left (5 \, x\right )\right ) + 2400 \, x^{2} \log \left (\log \left (5 \, x\right )\right )^{2} + 1280 \, x \log \left (\log \left (5 \, x\right )\right )^{3} + 256 \, \log \left (\log \left (5 \, x\right )\right )^{4} - 1500 \, x^{3} - 3600 \, x^{2} \log \left (\log \left (5 \, x\right )\right ) - 2880 \, x \log \left (\log \left (5 \, x\right )\right )^{2} - 768 \, \log \left (\log \left (5 \, x\right )\right )^{3} + 1350 \, x^{2} + 2160 \, x \log \left (\log \left (5 \, x\right )\right ) + 864 \, \log \left (\log \left (5 \, x\right )\right )^{2} - 541 \, x - 432 \, \log \left (\log \left (5 \, x\right )\right ) + 81\right ) - \frac {1}{5} \, \log \left (x\right ) \]

[In]

integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*log(5*x)-1024)*log(log(5*x))^3+((12000
*x^3-21600*x^2+10080*x-864)*log(5*x)-3840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*
log(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6755*x^2+1487*x-81)*log(5*x)-2000*x
^3+3600*x^2-2160*x+432)/(1280*x*log(5*x)*log(log(5*x))^4+(6400*x^2-3840*x)*log(5*x)*log(log(5*x))^3+(12000*x^3
-14400*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-2160*x)*log(5*x)*log(log(5*x))+(312
5*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*log(5*x)),x, algorithm="giac")

[Out]

x - 1/5*log(625*x^4 + 2000*x^3*log(log(5*x)) + 2400*x^2*log(log(5*x))^2 + 1280*x*log(log(5*x))^3 + 256*log(log
(5*x))^4 - 1500*x^3 - 3600*x^2*log(log(5*x)) - 2880*x*log(log(5*x))^2 - 768*log(log(5*x))^3 + 1350*x^2 + 2160*
x*log(log(5*x)) + 864*log(log(5*x))^2 - 541*x - 432*log(log(5*x)) + 81) - 1/5*log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=\int \frac {\ln \left (5\,x\right )\,\left (3125\,x^5-10625\,x^4+12750\,x^3-6755\,x^2+1487\,x-81\right )-2160\,x+3600\,x^2-2000\,x^3+{\ln \left (\ln \left (5\,x\right )\right )}^3\,\left (\ln \left (5\,x\right )\,\left (6400\,x^2-6400\,x+768\right )-1024\right )+{\ln \left (\ln \left (5\,x\right )\right )}^2\,\left (\ln \left (5\,x\right )\,\left (12000\,x^3-21600\,x^2+10080\,x-864\right )-3840\,x+2304\right )+\ln \left (\ln \left (5\,x\right )\right )\,\left (5760\,x+\ln \left (5\,x\right )\,\left (10000\,x^4-26000\,x^3+21600\,x^2-6480\,x+432\right )-4800\,x^2-1728\right )+\ln \left (5\,x\right )\,{\ln \left (\ln \left (5\,x\right )\right )}^4\,\left (1280\,x-256\right )+432}{1280\,x\,\ln \left (5\,x\right )\,{\ln \left (\ln \left (5\,x\right )\right )}^4-\ln \left (5\,x\right )\,\left (3840\,x-6400\,x^2\right )\,{\ln \left (\ln \left (5\,x\right )\right )}^3+\ln \left (5\,x\right )\,\left (12000\,x^3-14400\,x^2+4320\,x\right )\,{\ln \left (\ln \left (5\,x\right )\right )}^2-\ln \left (5\,x\right )\,\left (-10000\,x^4+18000\,x^3-10800\,x^2+2160\,x\right )\,\ln \left (\ln \left (5\,x\right )\right )+\ln \left (5\,x\right )\,\left (3125\,x^5-7500\,x^4+6750\,x^3-2705\,x^2+405\,x\right )} \,d x \]

[In]

int((log(5*x)*(1487*x - 6755*x^2 + 12750*x^3 - 10625*x^4 + 3125*x^5 - 81) - 2160*x + 3600*x^2 - 2000*x^3 + log
(log(5*x))^3*(log(5*x)*(6400*x^2 - 6400*x + 768) - 1024) + log(log(5*x))^2*(log(5*x)*(10080*x - 21600*x^2 + 12
000*x^3 - 864) - 3840*x + 2304) + log(log(5*x))*(5760*x + log(5*x)*(21600*x^2 - 6480*x - 26000*x^3 + 10000*x^4
+ 432) - 4800*x^2 - 1728) + log(5*x)*log(log(5*x))^4*(1280*x - 256) + 432)/(log(5*x)*(405*x - 2705*x^2 + 6750
*x^3 - 7500*x^4 + 3125*x^5) - log(5*x)*log(log(5*x))^3*(3840*x - 6400*x^2) + 1280*x*log(5*x)*log(log(5*x))^4 -
log(5*x)*log(log(5*x))*(2160*x - 10800*x^2 + 18000*x^3 - 10000*x^4) + log(5*x)*log(log(5*x))^2*(4320*x - 1440
0*x^2 + 12000*x^3)),x)

[Out]

int((log(5*x)*(1487*x - 6755*x^2 + 12750*x^3 - 10625*x^4 + 3125*x^5 - 81) - 2160*x + 3600*x^2 - 2000*x^3 + log
(log(5*x))^3*(log(5*x)*(6400*x^2 - 6400*x + 768) - 1024) + log(log(5*x))^2*(log(5*x)*(10080*x - 21600*x^2 + 12
000*x^3 - 864) - 3840*x + 2304) + log(log(5*x))*(5760*x + log(5*x)*(21600*x^2 - 6480*x - 26000*x^3 + 10000*x^4
+ 432) - 4800*x^2 - 1728) + log(5*x)*log(log(5*x))^4*(1280*x - 256) + 432)/(log(5*x)*(405*x - 2705*x^2 + 6750
*x^3 - 7500*x^4 + 3125*x^5) - log(5*x)*log(log(5*x))^3*(3840*x - 6400*x^2) + 1280*x*log(5*x)*log(log(5*x))^4 -
log(5*x)*log(log(5*x))*(2160*x - 10800*x^2 + 18000*x^3 - 10000*x^4) + log(5*x)*log(log(5*x))^2*(4320*x - 1440
0*x^2 + 12000*x^3)), x)