Integrand size = 204, antiderivative size = 25 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log \left (\log (2)+\log \left (9+\left (\frac {1}{1-\frac {2 x}{5}}+\frac {\log (x)}{2}\right )^2\right )\right ) \] Output:
ln(ln(2)+ln(9+(1/2*ln(x)+1/(-2/5*x+1))^2))
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right ) \] Input:
Integrate[(-500 - 80*x^2 + (-250 + 100*x - 40*x^2 + 16*x^3)*Log[x])/((-500 0*x + 5600*x^2 - 2160*x^3 + 288*x^4)*Log[2] + (-500*x + 400*x^2 - 80*x^3)* Log[2]*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[2]*Log[x]^2 + (-50 00*x + 5600*x^2 - 2160*x^3 + 288*x^4 + (-500*x + 400*x^2 - 80*x^3)*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[x]^2)*Log[(1000 - 720*x + 144*x^ 2 + (100 - 40*x)*Log[x] + (25 - 20*x + 4*x^2)*Log[x]^2)/(100 - 80*x + 16*x ^2)]),x]
Output:
Log[Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2 *x) + Log[x]^2/4]]
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.
| \(\displaystyle \int \frac {-80 x^2+\left (16 x^3-40 x^2+100 x-250\right ) \log (x)-500}{\left (-80 x^3+400 x^2-500 x\right ) \log (2) \log (x)+\left (8 x^4-60 x^3+150 x^2-125 x\right ) \log (2) \log ^2(x)+\left (288 x^4-2160 x^3+5600 x^2+\left (-80 x^3+400 x^2-500 x\right ) \log (x)+\left (8 x^4-60 x^3+150 x^2-125 x\right ) \log ^2(x)-5000 x\right ) \log \left (\frac {144 x^2+\left (4 x^2-20 x+25\right ) \log ^2(x)-720 x+(100-40 x) \log (x)+1000}{16 x^2-80 x+100}\right )+\left (288 x^4-2160 x^3+5600 x^2-5000 x\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (4 x^2+25\right ) (10-(2 x-5) \log (x))}{(5-2 x) x \left (8 \left (18 x^2-90 x+125\right )+(5-2 x)^2 \log ^2(x)-20 (2 x-5) \log (x)\right ) \left (\log \left (\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}+\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}\right )+\log (2)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (4 x^2+25\right ) ((5-2 x) \log (x)+10)}{(5-2 x) x \left ((5-2 x)^2 \log ^2(x)+20 (5-2 x) \log (x)+8 \left (18 x^2-90 x+125\right )\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (-\frac {5 (2 x \log (x)-5 \log (x)-10)}{x \left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}+\frac {20 (2 x \log (x)-5 \log (x)-10)}{(2 x-5) \left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}+\frac {2 (2 x \log (x)-5 \log (x)-10)}{\left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-20 \int \frac {1}{\left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx+50 \int \frac {1}{x \left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx-200 \int \frac {1}{(2 x-5) \left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx+25 \int \frac {\log (x)}{x \left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx+4 \int \frac {x \log (x)}{\left (4 \log ^2(x) x^2+144 x^2-20 \log ^2(x) x-40 \log (x) x-720 x+25 \log ^2(x)+100 \log (x)+1000\right ) \left (\log \left (\frac {\log ^2(x)}{4}+\frac {5 \log (x)}{5-2 x}+\frac {2 \left (18 x^2-90 x+125\right )}{(5-2 x)^2}\right )+\log (2)\right )}dx\right )\) |
Input:
Int[(-500 - 80*x^2 + (-250 + 100*x - 40*x^2 + 16*x^3)*Log[x])/((-5000*x + 5600*x^2 - 2160*x^3 + 288*x^4)*Log[2] + (-500*x + 400*x^2 - 80*x^3)*Log[2] *Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[2]*Log[x]^2 + (-5000*x + 5600*x^2 - 2160*x^3 + 288*x^4 + (-500*x + 400*x^2 - 80*x^3)*Log[x] + (-12 5*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[x]^2)*Log[(1000 - 720*x + 144*x^2 + (1 00 - 40*x)*Log[x] + (25 - 20*x + 4*x^2)*Log[x]^2)/(100 - 80*x + 16*x^2)]), x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(21)=42\).
Time = 37.61 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12
| method | result | size |
| parallelrisch | \(\ln \left (\ln \left (2\right )+\ln \left (\frac {\left (4 x^{2}-20 x +25\right ) \ln \left (x \right )^{2}+\left (-40 x +100\right ) \ln \left (x \right )+144 x^{2}-720 x +1000}{16 x^{2}-80 x +100}\right )\right )\) | \(53\) |
| default | \(\ln \left (\ln \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-\frac {5}{2}+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-\frac {5}{2}+x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )}{2}-i \pi \,\operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )\right ) \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )^{2}+\frac {i \pi \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x^{2}-5 x +\frac {25}{4}\right ) \ln \left (x \right )^{2}+\left (-10 x +25\right ) \ln \left (x \right )+36 x^{2}-180 x +250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{3}}{2}-\ln \left (2\right )-2 \ln \left (-\frac {5}{2}+x \right )\right )\) | \(374\) |
| risch | \(\ln \left (-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-\frac {5}{2}+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-\frac {5}{2}+x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )}{2}-i \pi \,\operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )\right ) \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )^{2}+\frac {i \pi \operatorname {csgn}\left (i \left (-\frac {5}{2}+x \right )^{2}\right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{2}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )}{\left (-\frac {5}{2}+x \right )^{2}}\right )}^{3}}{2}-\ln \left (2\right )-2 \ln \left (-\frac {5}{2}+x \right )+\ln \left (\left (\ln \left (x \right )^{2}+36\right ) x^{2}+\left (-5 \ln \left (x \right )^{2}-10 \ln \left (x \right )-180\right ) x +\frac {25 \ln \left (x \right )^{2}}{4}+25 \ln \left (x \right )+250\right )\right )\) | \(409\) |
Input:
int(((16*x^3-40*x^2+100*x-250)*ln(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2-1 25*x)*ln(x)^2+(-80*x^3+400*x^2-500*x)*ln(x)+288*x^4-2160*x^3+5600*x^2-5000 *x)*ln(((4*x^2-20*x+25)*ln(x)^2+(-40*x+100)*ln(x)+144*x^2-720*x+1000)/(16* x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*ln(2)*ln(x)^2+(-80*x^3+400*x^2 -500*x)*ln(2)*ln(x)+(288*x^4-2160*x^3+5600*x^2-5000*x)*ln(2)),x,method=_RE TURNVERBOSE)
Output:
ln(ln(2)+ln(1/4/(4*x^2-20*x+25)*((4*x^2-20*x+25)*ln(x)^2+(-40*x+100)*ln(x) +144*x^2-720*x+1000)))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log \left (\log \left (2\right ) + \log \left (\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right )^{2} + 144 \, x^{2} - 20 \, {\left (2 \, x - 5\right )} \log \left (x\right ) - 720 \, x + 1000}{4 \, {\left (4 \, x^{2} - 20 \, x + 25\right )}}\right )\right ) \] Input:
integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+15 0*x^2-125*x)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(x)+288*x^4-2160*x^3+5600 *x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-720* x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(- 80*x^3+400*x^2-500*x)*log(2)*log(x)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log (2)),x, algorithm="fricas")
Output:
log(log(2) + log(1/4*((4*x^2 - 20*x + 25)*log(x)^2 + 144*x^2 - 20*(2*x - 5 )*log(x) - 720*x + 1000)/(4*x^2 - 20*x + 25)))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 1.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log {\left (\log {\left (\frac {144 x^{2} - 720 x + \left (100 - 40 x\right ) \log {\left (x \right )} + \left (4 x^{2} - 20 x + 25\right ) \log {\left (x \right )}^{2} + 1000}{16 x^{2} - 80 x + 100} \right )} + \log {\left (2 \right )} \right )} \] Input:
integrate(((16*x**3-40*x**2+100*x-250)*ln(x)-80*x**2-500)/(((8*x**4-60*x** 3+150*x**2-125*x)*ln(x)**2+(-80*x**3+400*x**2-500*x)*ln(x)+288*x**4-2160*x **3+5600*x**2-5000*x)*ln(((4*x**2-20*x+25)*ln(x)**2+(-40*x+100)*ln(x)+144* x**2-720*x+1000)/(16*x**2-80*x+100))+(8*x**4-60*x**3+150*x**2-125*x)*ln(2) *ln(x)**2+(-80*x**3+400*x**2-500*x)*ln(2)*ln(x)+(288*x**4-2160*x**3+5600*x **2-5000*x)*ln(2)),x)
Output:
log(log((144*x**2 - 720*x + (100 - 40*x)*log(x) + (4*x**2 - 20*x + 25)*log (x)**2 + 1000)/(16*x**2 - 80*x + 100)) + log(2))
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log \left (-\log \left (2\right ) + \log \left ({\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right )^{2} + 144 \, x^{2} - 20 \, {\left (2 \, x - 5\right )} \log \left (x\right ) - 720 \, x + 1000\right ) - 2 \, \log \left (2 \, x - 5\right )\right ) \] Input:
integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+15 0*x^2-125*x)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(x)+288*x^4-2160*x^3+5600 *x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-720* x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(- 80*x^3+400*x^2-500*x)*log(2)*log(x)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log (2)),x, algorithm="maxima")
Output:
log(-log(2) + log((4*x^2 - 20*x + 25)*log(x)^2 + 144*x^2 - 20*(2*x - 5)*lo g(x) - 720*x + 1000) - 2*log(2*x - 5))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25) = 50\).
Time = 0.74 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\log \left (-\log \left (2\right ) + \log \left (4 \, x^{2} \log \left (x\right )^{2} - 20 \, x \log \left (x\right )^{2} + 144 \, x^{2} - 40 \, x \log \left (x\right ) + 25 \, \log \left (x\right )^{2} - 720 \, x + 100 \, \log \left (x\right ) + 1000\right ) - \log \left (4 \, x^{2} - 20 \, x + 25\right )\right ) \] Input:
integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+15 0*x^2-125*x)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(x)+288*x^4-2160*x^3+5600 *x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-720* x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(- 80*x^3+400*x^2-500*x)*log(2)*log(x)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log (2)),x, algorithm="giac")
Output:
log(-log(2) + log(4*x^2*log(x)^2 - 20*x*log(x)^2 + 144*x^2 - 40*x*log(x) + 25*log(x)^2 - 720*x + 100*log(x) + 1000) - log(4*x^2 - 20*x + 25))
Time = 4.80 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\ln \left (\ln \left (\frac {2\,\left ({\ln \left (x\right )}^2\,\left (4\,x^2-20\,x+25\right )-720\,x-\ln \left (x\right )\,\left (40\,x-100\right )+144\,x^2+1000\right )}{16\,x^2-80\,x+100}\right )\right ) \] Input:
int((80*x^2 - log(x)*(100*x - 40*x^2 + 16*x^3 - 250) + 500)/(log((log(x)^2 *(4*x^2 - 20*x + 25) - 720*x - log(x)*(40*x - 100) + 144*x^2 + 1000)/(16*x ^2 - 80*x + 100))*(5000*x + log(x)^2*(125*x - 150*x^2 + 60*x^3 - 8*x^4) - 5600*x^2 + 2160*x^3 - 288*x^4 + log(x)*(500*x - 400*x^2 + 80*x^3)) + log(2 )*(5000*x - 5600*x^2 + 2160*x^3 - 288*x^4) + log(2)*log(x)*(500*x - 400*x^ 2 + 80*x^3) + log(2)*log(x)^2*(125*x - 150*x^2 + 60*x^3 - 8*x^4)),x)
Output:
log(log((2*(log(x)^2*(4*x^2 - 20*x + 25) - 720*x - log(x)*(40*x - 100) + 1 44*x^2 + 1000))/(16*x^2 - 80*x + 100)))
\[ \int \frac {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx=\text {too large to display} \] Input:
int(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2- 125*x)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(x)+288*x^4-2160*x^3+5600*x^2-5 000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-720*x+1000 )/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(-80*x^3 +400*x^2-500*x)*log(2)*log(x)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log(2)),x )
Output:
2*( - 125*int(log(x)/(8*log((4*log(x)**2*x**2 - 20*log(x)**2*x + 25*log(x) **2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(16*x**2 - 80*x + 100))*log(x)**2*x**4 - 60*log((4*log(x)**2*x**2 - 20*log(x)**2*x + 25*lo g(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(16*x**2 - 8 0*x + 100))*log(x)**2*x**3 + 150*log((4*log(x)**2*x**2 - 20*log(x)**2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(16*x** 2 - 80*x + 100))*log(x)**2*x**2 - 125*log((4*log(x)**2*x**2 - 20*log(x)**2 *x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(1 6*x**2 - 80*x + 100))*log(x)**2*x - 80*log((4*log(x)**2*x**2 - 20*log(x)** 2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/( 16*x**2 - 80*x + 100))*log(x)*x**3 + 400*log((4*log(x)**2*x**2 - 20*log(x) **2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000) /(16*x**2 - 80*x + 100))*log(x)*x**2 - 500*log((4*log(x)**2*x**2 - 20*log( x)**2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 100 0)/(16*x**2 - 80*x + 100))*log(x)*x + 288*log((4*log(x)**2*x**2 - 20*log(x )**2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000 )/(16*x**2 - 80*x + 100))*x**4 - 2160*log((4*log(x)**2*x**2 - 20*log(x)**2 *x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(1 6*x**2 - 80*x + 100))*x**3 + 5600*log((4*log(x)**2*x**2 - 20*log(x)**2*x + 25*log(x)**2 - 40*log(x)*x + 100*log(x) + 144*x**2 - 720*x + 1000)/(16...