Integrand size = 144, antiderivative size = 31 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\left (2-\log \left (-x-x^2+\frac {x}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )\right )^2 \] Output:
(2-ln(x/ln(5*ln(3)/exp(2)/x)-x^2-x))^2
\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx \] Input:
Integrate[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2* x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x) ]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)]] )/(-(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2),x]
Output:
Integrate[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2* x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x) ]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)]] )/(-(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2), x]
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.
| \(\displaystyle \int \frac {\left ((4 x+2) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )-2\right ) \log \left (\frac {\left (-x^2-x\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )+x}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )+(-8 x-4) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+4}{\left (x^2+x\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )-x \log \left (\frac {5 \log (3)}{e^2 x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (2 x \log ^2\left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )-8 x \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )-4 \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )-\log \left (\frac {\log (243)}{x}\right )+8 x \left (1+\frac {1}{4} (\log (\log (243))-4) \log (\log (243))\right )+5 \left (1+\frac {1}{5} (\log (\log (243))-4) \log (\log (243))\right )\right ) \left (\log \left (x \left (-x+\frac {1}{\log \left (\frac {\log (243)}{x}\right )-2}-1\right )\right )-2\right )}{x \left (\log \left (\frac {1}{x}\right )-2 \left (1-\frac {1}{2} \log (\log (243))\right )\right ) \left (x \log \left (\frac {1}{x}\right )-2 x \left (1-\frac {1}{2} \log (\log (243))\right )+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (2 x \log ^2\left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )-4 x (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-2 (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-\log \left (\frac {\log (243)}{x}\right )+\log ^2(\log (243))-4 \log (\log (243))+2 x (2-\log (\log (243)))^2+5\right ) \left (2-\log \left (-x \left (x-\frac {1}{\log \left (\frac {\log (243)}{x}\right )-2}+1\right )\right )\right )}{x \left (-\log \left (\frac {1}{x}\right )-\log (\log (243))+2\right ) \left (-\log \left (\frac {1}{x}\right ) x+(2-\log (\log (243))) x-\log \left (\frac {1}{x}\right )-\log (\log (243))+3\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (2 x \log ^2\left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )-4 x (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-2 (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-\log \left (\frac {\log (243)}{x}\right )+\log ^2(\log (243))-4 \log (\log (243))+2 x (2-\log (\log (243)))^2+5\right ) \left (2-\log \left (-x \left (x+\frac {1}{2-\log \left (\frac {\log (243)}{x}\right )}+1\right )\right )\right )}{x \left (-\log \left (\frac {1}{x}\right )-\log (\log (243))+2\right ) \left (-\log \left (\frac {1}{x}\right ) x+(2-\log (\log (243))) x-\log \left (\frac {1}{x}\right )-\log (\log (243))+3\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -2 \int \frac {\left (2 x \log ^2\left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )-4 x (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-2 (2-\log (\log (243))) \log \left (\frac {1}{x}\right )-\log \left (\frac {\log (243)}{x}\right )+5 \left (1+\frac {1}{5} (-4+\log (\log (243))) \log (\log (243))\right )+2 x (2-\log (\log (243)))^2\right ) \left (2-\log \left (-x \left (x+\frac {1}{2-\log \left (\frac {\log (243)}{x}\right )}+1\right )\right )\right )}{x \left (\log \left (\frac {1}{x}\right )-2 \left (1-\frac {1}{2} \log (\log (243))\right )\right ) \left (\log \left (\frac {1}{x}\right ) x-(2-\log (\log (243))) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {2 \left (-2 x \log ^2\left (\frac {1}{x}\right )-\log ^2\left (\frac {1}{x}\right )+8 x \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )+4 \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )+\log \left (\frac {\log (243)}{x}\right )-8 x \left (1+\frac {1}{4} (-4+\log (\log (243))) \log (\log (243))\right )-5 \left (1+\frac {1}{5} (-4+\log (\log (243))) \log (\log (243))\right )\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}+\frac {\left (2 x \log ^2\left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )-8 x \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )-4 \left (1-\frac {1}{2} \log (\log (243))\right ) \log \left (\frac {1}{x}\right )-\log \left (\frac {\log (243)}{x}\right )+8 x \left (1+\frac {1}{4} (-4+\log (\log (243))) \log (\log (243))\right )+5 \left (1+\frac {1}{5} (-4+\log (\log (243))) \log (\log (243))\right )\right ) \log \left (-x^2-\frac {x}{2-\log \left (\frac {\log (243)}{x}\right )}-x\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (2 \int \frac {1}{x \left (-\log \left (\frac {1}{x}\right ) x+2 \left (1-\frac {1}{2} \log (\log (243))\right ) x-\log \left (\frac {1}{x}\right )+3 \left (1-\frac {1}{3} \log (\log (243))\right )\right )}dx-2 (1-\log (\log (243))) (3-\log (\log (243))) \int \frac {1}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+8 (2-\log (\log (243))) \int \frac {\log \left (\frac {1}{x}\right )}{\left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+4 (2-\log (\log (243))) \int \frac {\log \left (\frac {1}{x}\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+4 (2-\log (\log (243)))^2 \int \frac {1}{\left (-\log \left (\frac {1}{x}\right ) x+2 \left (1-\frac {1}{2} \log (\log (243))\right ) x-\log \left (\frac {1}{x}\right )+3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+4 \int \frac {\log ^2\left (\frac {1}{x}\right )}{\left (-\log \left (\frac {1}{x}\right ) x+2 \left (1-\frac {1}{2} \log (\log (243))\right ) x-\log \left (\frac {1}{x}\right )+3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+2 \int \frac {\log ^2\left (\frac {1}{x}\right )}{x \left (-\log \left (\frac {1}{x}\right ) x+2 \left (1-\frac {1}{2} \log (\log (243))\right ) x-\log \left (\frac {1}{x}\right )+3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+2 (2-\log (\log (243)))^2 \int \frac {\log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{\left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+\left (5-4 \log (\log (243))+\log ^2(\log (243))\right ) \int \frac {\log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx-4 (2-\log (\log (243))) \int \frac {\log \left (\frac {1}{x}\right ) \log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{\left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx-2 (2-\log (\log (243))) \int \frac {\log \left (\frac {1}{x}\right ) \log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+2 \int \frac {\log ^2\left (\frac {1}{x}\right ) \log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{\left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+\int \frac {\log ^2\left (\frac {1}{x}\right ) \log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{x \left (\log \left (\frac {1}{x}\right ) x-2 \left (1-\frac {1}{2} \log (\log (243))\right ) x+\log \left (\frac {1}{x}\right )-3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx+\int \frac {\log \left (\frac {\log (243)}{x}\right ) \log \left (-x^2+\frac {x}{\log \left (\frac {\log (243)}{x}\right )-2}-x\right )}{x \left (-\log \left (\frac {1}{x}\right ) x+2 \left (1-\frac {1}{2} \log (\log (243))\right ) x-\log \left (\frac {1}{x}\right )+3 \left (1-\frac {1}{3} \log (\log (243))\right )\right ) \left (2-\log \left (\frac {\log (243)}{x}\right )\right )}dx\right )\) |
Input:
Int[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2*x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x)]^2)*L og[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)]])/(-(x *Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(32)=64\).
Time = 2.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.81
| method | result | size |
| default | \(8 \ln \left (\frac {1}{x}\right )-4 \ln \left (\frac {\ln \left (5\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+\frac {\ln \left (\ln \left (3\right )\right )}{x}+\ln \left (5\right )+\ln \left (\frac {1}{x}\right )+\ln \left (\ln \left (3\right )\right )-\frac {3}{x}-2\right )+4 \ln \left (\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )\right )+\ln \left (\frac {x \left (-x \ln \left (5\right )-\ln \left (\ln \left (3\right )\right ) x +2 x -x \ln \left (\frac {1}{x}\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+3-\ln \left (\frac {1}{x}\right )\right )}{\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )}\right )^{2}\) | \(118\) |
| parts | \(8 \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )-4 \ln \left (\frac {5 \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right ) \ln \left (3\right )}{x}-\frac {5 \ln \left (3\right )}{x}+5 \ln \left (3\right ) \ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )\right )+4 \ln \left (\ln \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )\right )+\ln \left (\frac {x \left (-x \ln \left (5\right )-\ln \left (\ln \left (3\right )\right ) x +2 x -x \ln \left (\frac {1}{x}\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+3-\ln \left (\frac {1}{x}\right )\right )}{\ln \left (5\right )+\ln \left (\ln \left (3\right )\right )-2+\ln \left (\frac {1}{x}\right )}\right )^{2}\) | \(133\) |
Input:
int((((4*x+2)*ln(5*ln(3)/exp(2)/x)^2-2*ln(5*ln(3)/exp(2)/x)-2)*ln(((-x^2-x )*ln(5*ln(3)/exp(2)/x)+x)/ln(5*ln(3)/exp(2)/x))+(-8*x-4)*ln(5*ln(3)/exp(2) /x)^2+4*ln(5*ln(3)/exp(2)/x)+4)/((x^2+x)*ln(5*ln(3)/exp(2)/x)^2-x*ln(5*ln( 3)/exp(2)/x)),x,method=_RETURNVERBOSE)
Output:
8*ln(1/x)-4*ln(ln(5)/x+ln(1/x)/x+ln(ln(3))/x+ln(5)+ln(1/x)+ln(ln(3))-3/x-2 )+4*ln(ln(5)+ln(ln(3))-2+ln(1/x))+ln(x*(-x*ln(5)-ln(ln(3))*x+2*x-x*ln(1/x) -ln(5)-ln(ln(3))+3-ln(1/x))/(ln(5)+ln(ln(3))-2+ln(1/x)))^2
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (28) = 56\).
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.42 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right )^{2} - 4 \, \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) \] Input:
integrate((((2+4*x)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*l og(((-x^2-x)*log(5*log(3)/exp(2)/x)+x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*lo g(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)/e xp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="fricas")
Output:
log(-((x^2 + x)*log(5*e^(-2)*log(3)/x) - x)/log(5*e^(-2)*log(3)/x))^2 - 4* log(-((x^2 + x)*log(5*e^(-2)*log(3)/x) - x)/log(5*e^(-2)*log(3)/x))
Exception generated. \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((2+4*x)*ln(5*ln(3)/exp(2)/x)**2-2*ln(5*ln(3)/exp(2)/x)-2)*ln(( (-x**2-x)*ln(5*ln(3)/exp(2)/x)+x)/ln(5*ln(3)/exp(2)/x))+(-8*x-4)*ln(5*ln(3 )/exp(2)/x)**2+4*ln(5*ln(3)/exp(2)/x)+4)/((x**2+x)*ln(5*ln(3)/exp(2)/x)**2 -x*ln(5*ln(3)/exp(2)/x)),x)
Output:
Exception raised: PolynomialError >> 1/(x**3 + 2*x**2 + x) contains an ele ment of the set of generators.
\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int { -\frac {2 \, {\left (2 \, {\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - {\left ({\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 1\right )} \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) - 2 \, \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 2\right )}}{{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - x \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )} \,d x } \] Input:
integrate((((2+4*x)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*l og(((-x^2-x)*log(5*log(3)/exp(2)/x)+x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*lo g(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)/e xp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="maxima")
Output:
-2*integrate((2*(2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - ((2*x + 1)*log(5*e^(- 2)*log(3)/x)^2 - log(5*e^(-2)*log(3)/x) - 1)*log(-((x^2 + x)*log(5*e^(-2)* log(3)/x) - x)/log(5*e^(-2)*log(3)/x)) - 2*log(5*e^(-2)*log(3)/x) - 2)/((x ^2 + x)*log(5*e^(-2)*log(3)/x)^2 - x*log(5*e^(-2)*log(3)/x)), x)
\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=\int { -\frac {2 \, {\left (2 \, {\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - {\left ({\left (2 \, x + 1\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 1\right )} \log \left (-\frac {{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - x}{\log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )}\right ) - 2 \, \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right ) - 2\right )}}{{\left (x^{2} + x\right )} \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )^{2} - x \log \left (\frac {5 \, e^{\left (-2\right )} \log \left (3\right )}{x}\right )} \,d x } \] Input:
integrate((((2+4*x)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*l og(((-x^2-x)*log(5*log(3)/exp(2)/x)+x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*lo g(5*log(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)/e xp(2)/x)^2-x*log(5*log(3)/exp(2)/x)),x, algorithm="giac")
Output:
integrate(-2*(2*(2*x + 1)*log(5*e^(-2)*log(3)/x)^2 - ((2*x + 1)*log(5*e^(- 2)*log(3)/x)^2 - log(5*e^(-2)*log(3)/x) - 1)*log(-((x^2 + x)*log(5*e^(-2)* log(3)/x) - x)/log(5*e^(-2)*log(3)/x)) - 2*log(5*e^(-2)*log(3)/x) - 2)/((x ^2 + x)*log(5*e^(-2)*log(3)/x)^2 - x*log(5*e^(-2)*log(3)/x)), x)
Time = 1.92 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.19 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx={\ln \left (\frac {x-\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )\,\left (x^2+x\right )}{\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )}\right )}^2+4\,\ln \left (x+1\right )-4\,\ln \left (\frac {4\,\left (x-x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+1\right )}{x\,{\left (x+1\right )}^2}-\frac {4\,\left (2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-x+3\,x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+2\,x^2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-1\right )}{x\,{\left (x+1\right )}^2}\right )+4\,\ln \left (\frac {4\,\left (2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-x+3\,x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+2\,x^2\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )-1\right )}{x\,{\left (x+1\right )}^2}+\frac {4\,\left (x-x\,\ln \left (\frac {5\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x}\right )+1\right )}{x\,{\left (x+1\right )}^2}\right )-4\,\ln \left (x\,\left (x^2+x+1\right )\right ) \] Input:
int(-(4*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(8*x + 4) - log((x - log((5*exp(-2)*log(3))/x)*(x + x^2))/log((5*exp(-2)*log(3))/x)) *(2*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(4*x + 2) + 2) + 4)/(x*log((5*exp(-2)*log(3))/x) - log((5*exp(-2)*log(3))/x)^2*(x + x^2) ),x)
Output:
4*log(x + 1) - 4*log((4*(x - x*log((5*exp(-2)*log(3))/x) + 1))/(x*(x + 1)^ 2) - (4*(2*log((5*exp(-2)*log(3))/x) - x + 3*x*log((5*exp(-2)*log(3))/x) + 2*x^2*log((5*exp(-2)*log(3))/x) - 1))/(x*(x + 1)^2)) + 4*log((4*(2*log((5 *exp(-2)*log(3))/x) - x + 3*x*log((5*exp(-2)*log(3))/x) + 2*x^2*log((5*exp (-2)*log(3))/x) - 1))/(x*(x + 1)^2) + (4*(x - x*log((5*exp(-2)*log(3))/x) + 1))/(x*(x + 1)^2)) - 4*log(x*(x + x^2 + 1)) + log((x - log((5*exp(-2)*lo g(3))/x)*(x + x^2))/log((5*exp(-2)*log(3))/x))^2
Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13 \[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx=4 \,\mathrm {log}\left (\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right )\right )-4 \,\mathrm {log}\left (\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right ) x +\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right )-1\right )+{\mathrm {log}\left (\frac {-\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right ) x^{2}-\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right ) x +x}{\mathrm {log}\left (\frac {5 \,\mathrm {log}\left (3\right )}{e^{2} x}\right )}\right )}^{2}-4 \,\mathrm {log}\left (x \right ) \] Input:
int((((2+4*x)*log(5*log(3)/exp(2)/x)^2-2*log(5*log(3)/exp(2)/x)-2)*log(((- x^2-x)*log(5*log(3)/exp(2)/x)+x)/log(5*log(3)/exp(2)/x))+(-8*x-4)*log(5*lo g(3)/exp(2)/x)^2+4*log(5*log(3)/exp(2)/x)+4)/((x^2+x)*log(5*log(3)/exp(2)/ x)^2-x*log(5*log(3)/exp(2)/x)),x)
Output:
4*log(log((5*log(3))/(e**2*x))) - 4*log(log((5*log(3))/(e**2*x))*x + log(( 5*log(3))/(e**2*x)) - 1) + log(( - log((5*log(3))/(e**2*x))*x**2 - log((5* log(3))/(e**2*x))*x + x)/log((5*log(3))/(e**2*x)))**2 - 4*log(x)