Integrand size = 284, antiderivative size = 30 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x-\frac {1}{5} \log \left (\frac {1}{4} x \left (x-(-3+x+4 (x+\log (\log (5 x))))^4\right )\right ) \] Output:
x-1/5*ln(1/4*(x-(4*ln(ln(5*x))+5*x-3)^4)*x)
Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(30)=60\).
Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.37 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=\frac {1}{5} \left (5 x-\log (5 x)-\log \left (405-2705 x+6750 x^2-7500 x^3+3125 x^4-2160 \log (\log (5 x))+10800 x \log (\log (5 x))-18000 x^2 \log (\log (5 x))+10000 x^3 \log (\log (5 x))+4320 \log ^2(\log (5 x))-14400 x \log ^2(\log (5 x))+12000 x^2 \log ^2(\log (5 x))-3840 \log ^3(\log (5 x))+6400 x \log ^3(\log (5 x))+1280 \log ^4(\log (5 x))\right )\right ) \] Input:
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)*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])*Lo g[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 + 10 000*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]
Output:
(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 + 12000* 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
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(30)=60\).
Time = 10.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.30, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7292, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-2000 x^3+3600 x^2+\left (\left (6400 x^2-6400 x+768\right ) \log (5 x)-1024\right ) \log ^3(\log (5 x))+\left (\left (12000 x^3-21600 x^2+10080 x-864\right ) \log (5 x)-3840 x+2304\right ) \log ^2(\log (5 x))+\left (-4800 x^2+\left (10000 x^4-26000 x^3+21600 x^2-6480 x+432\right ) \log (5 x)+5760 x-1728\right ) \log (\log (5 x))+\left (3125 x^5-10625 x^4+12750 x^3-6755 x^2+1487 x-81\right ) \log (5 x)-2160 x+(1280 x-256) \log (5 x) \log ^4(\log (5 x))+432}{\left (6400 x^2-3840 x\right ) \log (5 x) \log ^3(\log (5 x))+\left (12000 x^3-14400 x^2+4320 x\right ) \log (5 x) \log ^2(\log (5 x))+\left (10000 x^4-18000 x^3+10800 x^2-2160 x\right ) \log (5 x) \log (\log (5 x))+\left (3125 x^5-7500 x^4+6750 x^3-2705 x^2+405 x\right ) \log (5 x)+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2000 x^3+3600 x^2+\left (\left (6400 x^2-6400 x+768\right ) \log (5 x)-1024\right ) \log ^3(\log (5 x))+\left (\left (12000 x^3-21600 x^2+10080 x-864\right ) \log (5 x)-3840 x+2304\right ) \log ^2(\log (5 x))+\left (-4800 x^2+\left (10000 x^4-26000 x^3+21600 x^2-6480 x+432\right ) \log (5 x)+5760 x-1728\right ) \log (\log (5 x))+\left (3125 x^5-10625 x^4+12750 x^3-6755 x^2+1487 x-81\right ) \log (5 x)-2160 x+(1280 x-256) \log (5 x) \log ^4(\log (5 x))+432}{5 x \log (5 x) \left (625 x^4-1500 x^3+2000 x^3 \log (\log (5 x))+1350 x^2+2400 x^2 \log ^2(\log (5 x))-3600 x^2 \log (\log (5 x))-541 x+256 \log ^4(\log (5 x))+1280 x \log ^3(\log (5 x))-768 \log ^3(\log (5 x))-2880 x \log ^2(\log (5 x))+864 \log ^2(\log (5 x))+2160 x \log (\log (5 x))-432 \log (\log (5 x))+81\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {-256 (1-5 x) \log (5 x) \log ^4(\log (5 x))-256 \left (4-\left (25 x^2-25 x+3\right ) \log (5 x)\right ) \log ^3(\log (5 x))+96 \left (-40 x-\left (-125 x^3+225 x^2-105 x+9\right ) \log (5 x)+24\right ) \log ^2(\log (5 x))-16 \left (300 x^2-360 x-\left (625 x^4-1625 x^3+1350 x^2-405 x+27\right ) \log (5 x)+108\right ) \log (\log (5 x))-2000 x^3+3600 x^2-2160 x-\left (-3125 x^5+10625 x^4-12750 x^3+6755 x^2-1487 x+81\right ) \log (5 x)+432}{x \log (5 x) \left (625 x^4+2000 \log (\log (5 x)) x^3-1500 x^3+2400 \log ^2(\log (5 x)) x^2-3600 \log (\log (5 x)) x^2+1350 x^2+1280 \log ^3(\log (5 x)) x-2880 \log ^2(\log (5 x)) x+2160 \log (\log (5 x)) x-541 x+256 \log ^4(\log (5 x))-768 \log ^3(\log (5 x))+864 \log ^2(\log (5 x))-432 \log (\log (5 x))+81\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {5 x-1}{x}+\frac {-2500 \log (5 x) x^4+4500 \log (5 x) x^3-6000 \log (5 x) \log (\log (5 x)) x^3-2000 x^3-4800 \log (5 x) \log ^2(\log (5 x)) x^2-2700 \log (5 x) x^2+7200 \log (5 x) \log (\log (5 x)) x^2-4800 \log (\log (5 x)) x^2+3600 x^2-1280 \log (5 x) \log ^3(\log (5 x)) x+2880 \log (5 x) \log ^2(\log (5 x)) x-3840 \log ^2(\log (5 x)) x+541 \log (5 x) x-2160 \log (5 x) \log (\log (5 x)) x+5760 \log (\log (5 x)) x-2160 x-1024 \log ^3(\log (5 x))+2304 \log ^2(\log (5 x))-1728 \log (\log (5 x))+432}{x \log (5 x) \left (625 x^4+2000 \log (\log (5 x)) x^3-1500 x^3+2400 \log ^2(\log (5 x)) x^2-3600 \log (\log (5 x)) x^2+1350 x^2+1280 \log ^3(\log (5 x)) x-2880 \log ^2(\log (5 x)) x+2160 \log (\log (5 x)) x-541 x+256 \log ^4(\log (5 x))-768 \log ^3(\log (5 x))+864 \log ^2(\log (5 x))-432 \log (\log (5 x))+81\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (-\log \left (625 x^4-1500 x^3+2000 x^3 \log (\log (5 x))+1350 x^2+2400 x^2 \log ^2(\log (5 x))-3600 x^2 \log (\log (5 x))-541 x+256 \log ^4(\log (5 x))+1280 x \log ^3(\log (5 x))-768 \log ^3(\log (5 x))-2880 x \log ^2(\log (5 x))+864 \log ^2(\log (5 x))+2160 x \log (\log (5 x))-432 \log (\log (5 x))+81\right )+5 x-\log (x)\right )\) |
Input:
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]] + (2 304 - 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 - 7 500*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]
Output:
(5*x - Log[x] - Log[81 - 541*x + 1350*x^2 - 1500*x^3 + 625*x^4 - 432*Log[L og[5*x]] + 2160*x*Log[Log[5*x]] - 3600*x^2*Log[Log[5*x]] + 2000*x^3*Log[Lo g[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
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(26)=52\).
Time = 2.04 (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 | \(x -\frac {\ln \left (5 x \right )}{5}-\frac {\ln \left (x^{4}+\frac {16 \ln \left (\ln \left (5 x \right )\right ) x^{3}}{5}+\frac {96 \ln \left (\ln \left (5 x \right )\right )^{2} x^{2}}{25}+\frac {256 \ln \left (\ln \left (5 x \right )\right )^{3} x}{125}+\frac {256 \ln \left (\ln \left (5 x \right )\right )^{4}}{625}-\frac {12 x^{3}}{5}-\frac {144 \ln \left (\ln \left (5 x \right )\right ) x^{2}}{25}-\frac {576 \ln \left (\ln \left (5 x \right )\right )^{2} x}{125}-\frac {768 \ln \left (\ln \left (5 x \right )\right )^{3}}{625}+\frac {54 x^{2}}{25}+\frac {432 \ln \left (\ln \left (5 x \right )\right ) x}{125}+\frac {864 \ln \left (\ln \left (5 x \right )\right )^{2}}{625}-\frac {541 x}{625}-\frac {432 \ln \left (\ln \left (5 x \right )\right )}{625}+\frac {81}{625}\right )}{5}\) | \(124\) |
Input:
int(((1280*x-256)*ln(5*x)*ln(ln(5*x))^4+((6400*x^2-6400*x+768)*ln(5*x)-102 4)*ln(ln(5*x))^3+((12000*x^3-21600*x^2+10080*x-864)*ln(5*x)-3840*x+2304)*l n(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))+(31 25*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*ln(5*x)),x,method=_RETURNVERBOSE)
Output:
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))+625/2 56*x^4-375/64*x^3+675/128*x^2-541/256*x+81/256)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (26) = 52\).
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (625 \, x^{4} + 256 \, {\left (5 \, x - 3\right )} \log \left (\log \left (5 \, x\right )\right )^{3} + 256 \, \log \left (\log \left (5 \, x\right )\right )^{4} - 1500 \, x^{3} + 96 \, {\left (25 \, x^{2} - 30 \, x + 9\right )} \log \left (\log \left (5 \, x\right )\right )^{2} + 1350 \, x^{2} + 16 \, {\left (125 \, x^{3} - 225 \, x^{2} + 135 \, x - 27\right )} \log \left (\log \left (5 \, x\right )\right ) - 541 \, x + 81\right ) - \frac {1}{5} \, \log \left (5 \, x\right ) \] Input:
integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*lo g(5*x)-1024)*log(log(5*x))^3+((12000*x^3-21600*x^2+10080*x-864)*log(5*x)-3 840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*lo g(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6 755*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^3-144 00*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-216 0*x)*log(5*x)*log(log(5*x))+(3125*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*lo g(5*x)),x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {\log {\left (x \right )}}{5} - \frac {\log {\left (\frac {625 x^{4}}{256} - \frac {375 x^{3}}{64} + \frac {675 x^{2}}{128} - \frac {541 x}{256} + \left (5 x - 3\right ) \log {\left (\log {\left (5 x \right )} \right )}^{3} + \left (\frac {75 x^{2}}{8} - \frac {45 x}{4} + \frac {27}{8}\right ) \log {\left (\log {\left (5 x \right )} \right )}^{2} + \left (\frac {125 x^{3}}{16} - \frac {225 x^{2}}{16} + \frac {135 x}{16} - \frac {27}{16}\right ) \log {\left (\log {\left (5 x \right )} \right )} + \log {\left (\log {\left (5 x \right )} \right )}^{4} + \frac {81}{256} \right )}}{5} \] Input:
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)-384 0*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**3-14 400*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)
Output:
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(log( 5*x))**4 + 81/256)/5
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.13 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (\frac {625}{256} \, x^{4} + {\left (5 \, x - 3\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{3} + \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{4} - \frac {375}{64} \, x^{3} + \frac {3}{8} \, {\left (25 \, x^{2} - 30 \, x + 9\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right )^{2} + \frac {675}{128} \, x^{2} + \frac {1}{16} \, {\left (125 \, x^{3} - 225 \, x^{2} + 135 \, x - 27\right )} \log \left (\log \left (5\right ) + \log \left (x\right )\right ) - \frac {541}{256} \, x + \frac {81}{256}\right ) - \frac {1}{5} \, \log \left (x\right ) \] Input:
integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*lo g(5*x)-1024)*log(log(5*x))^3+((12000*x^3-21600*x^2+10080*x-864)*log(5*x)-3 840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*lo g(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6 755*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^3-144 00*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-216 0*x)*log(5*x)*log(log(5*x))+(3125*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*lo g(5*x)),x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.10 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=x - \frac {1}{5} \, \log \left (625 \, x^{4} + 2000 \, x^{3} \log \left (\log \left (5 \, x\right )\right ) + 2400 \, x^{2} \log \left (\log \left (5 \, x\right )\right )^{2} + 1280 \, x \log \left (\log \left (5 \, x\right )\right )^{3} + 256 \, \log \left (\log \left (5 \, x\right )\right )^{4} - 1500 \, x^{3} - 3600 \, x^{2} \log \left (\log \left (5 \, x\right )\right ) - 2880 \, x \log \left (\log \left (5 \, x\right )\right )^{2} - 768 \, \log \left (\log \left (5 \, x\right )\right )^{3} + 1350 \, x^{2} + 2160 \, x \log \left (\log \left (5 \, x\right )\right ) + 864 \, \log \left (\log \left (5 \, x\right )\right )^{2} - 541 \, x - 432 \, \log \left (\log \left (5 \, x\right )\right ) + 81\right ) - \frac {1}{5} \, \log \left (x\right ) \] Input:
integrate(((1280*x-256)*log(5*x)*log(log(5*x))^4+((6400*x^2-6400*x+768)*lo g(5*x)-1024)*log(log(5*x))^3+((12000*x^3-21600*x^2+10080*x-864)*log(5*x)-3 840*x+2304)*log(log(5*x))^2+((10000*x^4-26000*x^3+21600*x^2-6480*x+432)*lo g(5*x)-4800*x^2+5760*x-1728)*log(log(5*x))+(3125*x^5-10625*x^4+12750*x^3-6 755*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^3-144 00*x^2+4320*x)*log(5*x)*log(log(5*x))^2+(10000*x^4-18000*x^3+10800*x^2-216 0*x)*log(5*x)*log(log(5*x))+(3125*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*lo g(5*x)),x, algorithm="giac")
Output:
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)
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 \] Input:
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 - 640 0*x + 768) - 1024) + log(log(5*x))^2*(log(5*x)*(10080*x - 21600*x^2 + 1200 0*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)*lo g(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) + 1 280*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 - 14400*x^2 + 12000*x^3)),x)
Output:
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 - 640 0*x + 768) - 1024) + log(log(5*x))^2*(log(5*x)*(10080*x - 21600*x^2 + 1200 0*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)*lo g(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) + 1 280*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 - 14400*x^2 + 12000*x^3)), x)
Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.17 \[ \int \frac {432-2160 x+3600 x^2-2000 x^3+\left (-81+1487 x-6755 x^2+12750 x^3-10625 x^4+3125 x^5\right ) \log (5 x)+\left (-1728+5760 x-4800 x^2+\left (432-6480 x+21600 x^2-26000 x^3+10000 x^4\right ) \log (5 x)\right ) \log (\log (5 x))+\left (2304-3840 x+\left (-864+10080 x-21600 x^2+12000 x^3\right ) \log (5 x)\right ) \log ^2(\log (5 x))+\left (-1024+\left (768-6400 x+6400 x^2\right ) \log (5 x)\right ) \log ^3(\log (5 x))+(-256+1280 x) \log (5 x) \log ^4(\log (5 x))}{\left (405 x-2705 x^2+6750 x^3-7500 x^4+3125 x^5\right ) \log (5 x)+\left (-2160 x+10800 x^2-18000 x^3+10000 x^4\right ) \log (5 x) \log (\log (5 x))+\left (4320 x-14400 x^2+12000 x^3\right ) \log (5 x) \log ^2(\log (5 x))+\left (-3840 x+6400 x^2\right ) \log (5 x) \log ^3(\log (5 x))+1280 x \log (5 x) \log ^4(\log (5 x))} \, dx=-\frac {\mathrm {log}\left (256 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{4}+1280 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{3} x -768 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{3}+2400 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{2} x^{2}-2880 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{2} x +864 \mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )^{2}+2000 \,\mathrm {log}\left (\mathrm {log}\left (5 x \right )\right ) x^{3}-3600 \,\mathrm {log}\left (\mathrm {log}\left (5 x \right )\right ) x^{2}+2160 \,\mathrm {log}\left (\mathrm {log}\left (5 x \right )\right ) x -432 \,\mathrm {log}\left (\mathrm {log}\left (5 x \right )\right )+625 x^{4}-1500 x^{3}+1350 x^{2}-541 x +81\right )}{5}-\frac {\mathrm {log}\left (5 x \right )}{5}+x \] Input:
int(((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(l og(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)*l og(5*x)*log(log(5*x))+(3125*x^5-7500*x^4+6750*x^3-2705*x^2+405*x)*log(5*x) ),x)
Output:
( - log(256*log(log(5*x))**4 + 1280*log(log(5*x))**3*x - 768*log(log(5*x)) **3 + 2400*log(log(5*x))**2*x**2 - 2880*log(log(5*x))**2*x + 864*log(log(5 *x))**2 + 2000*log(log(5*x))*x**3 - 3600*log(log(5*x))*x**2 + 2160*log(log (5*x))*x - 432*log(log(5*x)) + 625*x**4 - 1500*x**3 + 1350*x**2 - 541*x + 81) - log(5*x) + 5*x)/5