Integrand size = 326, antiderivative size = 28 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=e^4+x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \] Output:
x/(ln(2+1/ln(1/x))-ln(x)+2*x)+exp(4)+x^2
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \] Input:
Integrate[(-1 + (1 + 8*x^3)*Log[x^(-1)] + (2 + 16*x^3)*Log[x^(-1)]^2 + ((- 1 - 8*x^2)*Log[x^(-1)] + (-2 - 16*x^2)*Log[x^(-1)]^2)*Log[x] + (2*x*Log[x^ (-1)] + 4*x*Log[x^(-1)]^2)*Log[x]^2 + ((1 + 8*x^2)*Log[x^(-1)] + (2 + 16*x ^2)*Log[x^(-1)]^2 + (-4*x*Log[x^(-1)] - 8*x*Log[x^(-1)]^2)*Log[x])*Log[(1 + 2*Log[x^(-1)])/Log[x^(-1)]] + (2*x*Log[x^(-1)] + 4*x*Log[x^(-1)]^2)*Log[ (1 + 2*Log[x^(-1)])/Log[x^(-1)]]^2)/(4*x^2*Log[x^(-1)] + 8*x^2*Log[x^(-1)] ^2 + (-4*x*Log[x^(-1)] - 8*x*Log[x^(-1)]^2)*Log[x] + (Log[x^(-1)] + 2*Log[ x^(-1)]^2)*Log[x]^2 + (4*x*Log[x^(-1)] + 8*x*Log[x^(-1)]^2 + (-2*Log[x^(-1 )] - 4*Log[x^(-1)]^2)*Log[x])*Log[(1 + 2*Log[x^(-1)])/Log[x^(-1)]] + (Log[ x^(-1)] + 2*Log[x^(-1)]^2)*Log[(1 + 2*Log[x^(-1)])/Log[x^(-1)]]^2),x]
Output:
x^2 + x/(2*x - Log[x] + Log[2 + Log[x^(-1)]^(-1)])
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.
| \(\displaystyle \int \frac {\left (16 x^3+2\right ) \log ^2\left (\frac {1}{x}\right )+\left (8 x^3+1\right ) \log \left (\frac {1}{x}\right )+\left (\left (-16 x^2-2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-8 x^2-1\right ) \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (\left (16 x^2+2\right ) \log ^2\left (\frac {1}{x}\right )+\left (8 x^2+1\right ) \log \left (\frac {1}{x}\right )+\left (-8 x \log ^2\left (\frac {1}{x}\right )-4 x \log \left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {2 \log \left (\frac {1}{x}\right )+1}{\log \left (\frac {1}{x}\right )}\right )+\left (4 x \log ^2\left (\frac {1}{x}\right )+2 x \log \left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log ^2\left (\frac {1}{x}\right )+2 x \log \left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {2 \log \left (\frac {1}{x}\right )+1}{\log \left (\frac {1}{x}\right )}\right )-1}{8 x^2 \log ^2\left (\frac {1}{x}\right )+4 x^2 \log \left (\frac {1}{x}\right )+\left (2 \log ^2\left (\frac {1}{x}\right )+\log \left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (2 \log ^2\left (\frac {1}{x}\right )+\log \left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {2 \log \left (\frac {1}{x}\right )+1}{\log \left (\frac {1}{x}\right )}\right )+\left (-8 x \log ^2\left (\frac {1}{x}\right )-4 x \log \left (\frac {1}{x}\right )\right ) \log (x)+\left (8 x \log ^2\left (\frac {1}{x}\right )+\left (-4 \log ^2\left (\frac {1}{x}\right )-2 \log \left (\frac {1}{x}\right )\right ) \log (x)+4 x \log \left (\frac {1}{x}\right )\right ) \log \left (\frac {2 \log \left (\frac {1}{x}\right )+1}{\log \left (\frac {1}{x}\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (8 x^3+\left (8 x^2+1\right ) \log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )-\log (x) \left (8 x^2+4 x \log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )+1\right )+2 x \log ^2(x)+2 x \log ^2\left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )+1\right ) \log ^2\left (\frac {1}{x}\right )+\left (8 x^3+\left (8 x^2+1\right ) \log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )-\log (x) \left (8 x^2+4 x \log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )+1\right )+2 x \log ^2(x)+2 x \log ^2\left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )+1\right ) \log \left (\frac {1}{x}\right )-1}{\log \left (\frac {1}{x}\right ) \left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x+\frac {-4 x \log ^2\left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )-2 x \log \left (\frac {1}{x}\right )+\log \left (\frac {1}{x}\right )-1}{\log \left (\frac {1}{x}\right ) \left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )\right )^2}+\frac {1}{2 x-\log (x)+\log \left (\frac {1}{\log \left (\frac {1}{x}\right )}+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}dx-2 \int \frac {x}{\left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}dx-\int \frac {1}{\log \left (\frac {1}{x}\right ) \left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}dx+2 \int \frac {\log \left (\frac {1}{x}\right )}{\left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}dx-4 \int \frac {x \log \left (\frac {1}{x}\right )}{\left (2 \log \left (\frac {1}{x}\right )+1\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}dx+\int \frac {1}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )}dx+x^2\) |
Input:
Int[(-1 + (1 + 8*x^3)*Log[x^(-1)] + (2 + 16*x^3)*Log[x^(-1)]^2 + ((-1 - 8* x^2)*Log[x^(-1)] + (-2 - 16*x^2)*Log[x^(-1)]^2)*Log[x] + (2*x*Log[x^(-1)] + 4*x*Log[x^(-1)]^2)*Log[x]^2 + ((1 + 8*x^2)*Log[x^(-1)] + (2 + 16*x^2)*Lo g[x^(-1)]^2 + (-4*x*Log[x^(-1)] - 8*x*Log[x^(-1)]^2)*Log[x])*Log[(1 + 2*Lo g[x^(-1)])/Log[x^(-1)]] + (2*x*Log[x^(-1)] + 4*x*Log[x^(-1)]^2)*Log[(1 + 2 *Log[x^(-1)])/Log[x^(-1)]]^2)/(4*x^2*Log[x^(-1)] + 8*x^2*Log[x^(-1)]^2 + ( -4*x*Log[x^(-1)] - 8*x*Log[x^(-1)]^2)*Log[x] + (Log[x^(-1)] + 2*Log[x^(-1) ]^2)*Log[x]^2 + (4*x*Log[x^(-1)] + 8*x*Log[x^(-1)]^2 + (-2*Log[x^(-1)] - 4 *Log[x^(-1)]^2)*Log[x])*Log[(1 + 2*Log[x^(-1)])/Log[x^(-1)]] + (Log[x^(-1) ] + 2*Log[x^(-1)]^2)*Log[(1 + 2*Log[x^(-1)])/Log[x^(-1)]]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(27)=54\).
Time = 9.99 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04
| method | result | size |
| parallelrisch | \(\frac {-48 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )-96 x^{2} \ln \left (x \right )+48 \ln \left (x \right )+192 x^{3}+96 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right ) x^{2}}{192 x -96 \ln \left (x \right )+96 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}\) | \(85\) |
| risch | \(x^{2}+\frac {2 x}{-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{3}+2 \ln \left (2\right )+4 x -2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\ln \left (x \right )-\frac {1}{2}\right )}\) | \(136\) |
| derivativedivides | \(x^{2}+\frac {2 i}{\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{3}}{x}+\frac {2 i \ln \left (2\right )}{x}-\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )\right )}{x}+\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{x}+\frac {2 i \ln \left (\frac {1}{x}\right )}{x}+4 i}\) | \(187\) |
| default | \(x^{2}+\frac {2 i}{\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{3}}{x}+\frac {2 i \ln \left (2\right )}{x}-\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )\right )}{x}+\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{x}+\frac {2 i \ln \left (\frac {1}{x}\right )}{x}+4 i}\) | \(187\) |
Input:
int(((4*x*ln(1/x)^2+2*x*ln(1/x))*ln((2*ln(1/x)+1)/ln(1/x))^2+((-8*x*ln(1/x )^2-4*x*ln(1/x))*ln(x)+(16*x^2+2)*ln(1/x)^2+(8*x^2+1)*ln(1/x))*ln((2*ln(1/ x)+1)/ln(1/x))+(4*x*ln(1/x)^2+2*x*ln(1/x))*ln(x)^2+((-16*x^2-2)*ln(1/x)^2+ (-8*x^2-1)*ln(1/x))*ln(x)+(16*x^3+2)*ln(1/x)^2+(8*x^3+1)*ln(1/x)-1)/((2*ln (1/x)^2+ln(1/x))*ln((2*ln(1/x)+1)/ln(1/x))^2+((-4*ln(1/x)^2-2*ln(1/x))*ln( x)+8*x*ln(1/x)^2+4*x*ln(1/x))*ln((2*ln(1/x)+1)/ln(1/x))+(2*ln(1/x)^2+ln(1/ x))*ln(x)^2+(-8*x*ln(1/x)^2-4*x*ln(1/x))*ln(x)+8*x^2*ln(1/x)^2+4*x^2*ln(1/ x)),x,method=_RETURNVERBOSE)
Output:
1/96*(-48*ln((2*ln(1/x)+1)/ln(1/x))-96*x^2*ln(x)+48*ln(x)+192*x^3+96*ln((2 *ln(1/x)+1)/ln(1/x))*x^2)/(2*x-ln(x)+ln((2*ln(1/x)+1)/ln(1/x)))
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\frac {2 \, x^{3} + x^{2} \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + x^{2} \log \left (\frac {1}{x}\right ) + x}{2 \, x + \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + \log \left (\frac {1}{x}\right )} \] Input:
integrate(((4*x*log(1/x)^2+2*x*log(1/x))*log((2*log(1/x)+1)/log(1/x))^2+(( -8*x*log(1/x)^2-4*x*log(1/x))*log(x)+(16*x^2+2)*log(1/x)^2+(8*x^2+1)*log(1 /x))*log((2*log(1/x)+1)/log(1/x))+(4*x*log(1/x)^2+2*x*log(1/x))*log(x)^2+( (-16*x^2-2)*log(1/x)^2+(-8*x^2-1)*log(1/x))*log(x)+(16*x^3+2)*log(1/x)^2+( 8*x^3+1)*log(1/x)-1)/((2*log(1/x)^2+log(1/x))*log((2*log(1/x)+1)/log(1/x)) ^2+((-4*log(1/x)^2-2*log(1/x))*log(x)+8*x*log(1/x)^2+4*x*log(1/x))*log((2* log(1/x)+1)/log(1/x))+(2*log(1/x)^2+log(1/x))*log(x)^2+(-8*x*log(1/x)^2-4* x*log(1/x))*log(x)+8*x^2*log(1/x)^2+4*x^2*log(1/x)),x, algorithm="fricas")
Output:
(2*x^3 + x^2*log((2*log(1/x) + 1)/log(1/x)) + x^2*log(1/x) + x)/(2*x + log ((2*log(1/x) + 1)/log(1/x)) + log(1/x))
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^{2} + \frac {x}{2 x - \log {\left (x \right )} + \log {\left (- \frac {1 - 2 \log {\left (x \right )}}{\log {\left (x \right )}} \right )}} \] Input:
integrate(((4*x*ln(1/x)**2+2*x*ln(1/x))*ln((2*ln(1/x)+1)/ln(1/x))**2+((-8* x*ln(1/x)**2-4*x*ln(1/x))*ln(x)+(16*x**2+2)*ln(1/x)**2+(8*x**2+1)*ln(1/x)) *ln((2*ln(1/x)+1)/ln(1/x))+(4*x*ln(1/x)**2+2*x*ln(1/x))*ln(x)**2+((-16*x** 2-2)*ln(1/x)**2+(-8*x**2-1)*ln(1/x))*ln(x)+(16*x**3+2)*ln(1/x)**2+(8*x**3+ 1)*ln(1/x)-1)/((2*ln(1/x)**2+ln(1/x))*ln((2*ln(1/x)+1)/ln(1/x))**2+((-4*ln (1/x)**2-2*ln(1/x))*ln(x)+8*x*ln(1/x)**2+4*x*ln(1/x))*ln((2*ln(1/x)+1)/ln( 1/x))+(2*ln(1/x)**2+ln(1/x))*ln(x)**2+(-8*x*ln(1/x)**2-4*x*ln(1/x))*ln(x)+ 8*x**2*ln(1/x)**2+4*x**2*ln(1/x)),x)
Output:
x**2 + x/(2*x - log(x) + log(-(1 - 2*log(x))/log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\frac {2 \, x^{3} - x^{2} \log \left (x\right ) + x^{2} \log \left (2 \, \log \left (x\right ) - 1\right ) - x^{2} \log \left (\log \left (x\right )\right ) + x}{2 \, x - \log \left (x\right ) + \log \left (2 \, \log \left (x\right ) - 1\right ) - \log \left (\log \left (x\right )\right )} \] Input:
integrate(((4*x*log(1/x)^2+2*x*log(1/x))*log((2*log(1/x)+1)/log(1/x))^2+(( -8*x*log(1/x)^2-4*x*log(1/x))*log(x)+(16*x^2+2)*log(1/x)^2+(8*x^2+1)*log(1 /x))*log((2*log(1/x)+1)/log(1/x))+(4*x*log(1/x)^2+2*x*log(1/x))*log(x)^2+( (-16*x^2-2)*log(1/x)^2+(-8*x^2-1)*log(1/x))*log(x)+(16*x^3+2)*log(1/x)^2+( 8*x^3+1)*log(1/x)-1)/((2*log(1/x)^2+log(1/x))*log((2*log(1/x)+1)/log(1/x)) ^2+((-4*log(1/x)^2-2*log(1/x))*log(x)+8*x*log(1/x)^2+4*x*log(1/x))*log((2* log(1/x)+1)/log(1/x))+(2*log(1/x)^2+log(1/x))*log(x)^2+(-8*x*log(1/x)^2-4* x*log(1/x))*log(x)+8*x^2*log(1/x)^2+4*x^2*log(1/x)),x, algorithm="maxima")
Output:
(2*x^3 - x^2*log(x) + x^2*log(2*log(x) - 1) - x^2*log(log(x)) + x)/(2*x - log(x) + log(2*log(x) - 1) - log(log(x)))
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^{2} + \frac {x}{2 \, x - \log \left (x\right ) + \log \left (2 \, \log \left (x\right ) - 1\right ) - \log \left (\log \left (x\right )\right )} \] Input:
integrate(((4*x*log(1/x)^2+2*x*log(1/x))*log((2*log(1/x)+1)/log(1/x))^2+(( -8*x*log(1/x)^2-4*x*log(1/x))*log(x)+(16*x^2+2)*log(1/x)^2+(8*x^2+1)*log(1 /x))*log((2*log(1/x)+1)/log(1/x))+(4*x*log(1/x)^2+2*x*log(1/x))*log(x)^2+( (-16*x^2-2)*log(1/x)^2+(-8*x^2-1)*log(1/x))*log(x)+(16*x^3+2)*log(1/x)^2+( 8*x^3+1)*log(1/x)-1)/((2*log(1/x)^2+log(1/x))*log((2*log(1/x)+1)/log(1/x)) ^2+((-4*log(1/x)^2-2*log(1/x))*log(x)+8*x*log(1/x)^2+4*x*log(1/x))*log((2* log(1/x)+1)/log(1/x))+(2*log(1/x)^2+log(1/x))*log(x)^2+(-8*x*log(1/x)^2-4* x*log(1/x))*log(x)+8*x^2*log(1/x)^2+4*x^2*log(1/x)),x, algorithm="giac")
Output:
x^2 + x/(2*x - log(x) + log(2*log(x) - 1) - log(log(x)))
Timed out. \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\int \frac {{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+\ln \left (\frac {1}{x}\right )\,\left (8\,x^3+1\right )+\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (\ln \left (\frac {1}{x}\right )\,\left (8\,x^2+1\right )-\ln \left (x\right )\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^2+2\right )\right )-\ln \left (x\right )\,\left (\left (16\,x^2+2\right )\,{\ln \left (\frac {1}{x}\right )}^2+\left (8\,x^2+1\right )\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (x\right )}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^3+2\right )-1}{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (4\,x\,\ln \left (\frac {1}{x}\right )-\ln \left (x\right )\,\left (4\,{\ln \left (\frac {1}{x}\right )}^2+2\,\ln \left (\frac {1}{x}\right )\right )+8\,x\,{\ln \left (\frac {1}{x}\right )}^2\right )-\ln \left (x\right )\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (x\right )}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+4\,x^2\,\ln \left (\frac {1}{x}\right )+{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+8\,x^2\,{\ln \left (\frac {1}{x}\right )}^2} \,d x \] Input:
int((log((2*log(1/x) + 1)/log(1/x))^2*(2*x*log(1/x) + 4*x*log(1/x)^2) + lo g(1/x)*(8*x^3 + 1) + log((2*log(1/x) + 1)/log(1/x))*(log(1/x)*(8*x^2 + 1) - log(x)*(4*x*log(1/x) + 8*x*log(1/x)^2) + log(1/x)^2*(16*x^2 + 2)) - log( x)*(log(1/x)*(8*x^2 + 1) + log(1/x)^2*(16*x^2 + 2)) + log(x)^2*(2*x*log(1/ x) + 4*x*log(1/x)^2) + log(1/x)^2*(16*x^3 + 2) - 1)/(log((2*log(1/x) + 1)/ log(1/x))*(4*x*log(1/x) - log(x)*(2*log(1/x) + 4*log(1/x)^2) + 8*x*log(1/x )^2) - log(x)*(4*x*log(1/x) + 8*x*log(1/x)^2) + log(x)^2*(log(1/x) + 2*log (1/x)^2) + 4*x^2*log(1/x) + log((2*log(1/x) + 1)/log(1/x))^2*(log(1/x) + 2 *log(1/x)^2) + 8*x^2*log(1/x)^2),x)
Output:
int((log((2*log(1/x) + 1)/log(1/x))^2*(2*x*log(1/x) + 4*x*log(1/x)^2) + lo g(1/x)*(8*x^3 + 1) + log((2*log(1/x) + 1)/log(1/x))*(log(1/x)*(8*x^2 + 1) - log(x)*(4*x*log(1/x) + 8*x*log(1/x)^2) + log(1/x)^2*(16*x^2 + 2)) - log( x)*(log(1/x)*(8*x^2 + 1) + log(1/x)^2*(16*x^2 + 2)) + log(x)^2*(2*x*log(1/ x) + 4*x*log(1/x)^2) + log(1/x)^2*(16*x^3 + 2) - 1)/(log((2*log(1/x) + 1)/ log(1/x))*(4*x*log(1/x) - log(x)*(2*log(1/x) + 4*log(1/x)^2) + 8*x*log(1/x )^2) - log(x)*(4*x*log(1/x) + 8*x*log(1/x)^2) + log(x)^2*(log(1/x) + 2*log (1/x)^2) + 4*x^2*log(1/x) + log((2*log(1/x) + 1)/log(1/x))^2*(log(1/x) + 2 *log(1/x)^2) + 8*x^2*log(1/x)^2), x)
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\frac {2 \,\mathrm {log}\left (\frac {2 \,\mathrm {log}\left (x \right )-1}{\mathrm {log}\left (x \right )}\right ) x^{2}-\mathrm {log}\left (\frac {2 \,\mathrm {log}\left (x \right )-1}{\mathrm {log}\left (x \right )}\right )-2 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right )+4 x^{3}}{2 \,\mathrm {log}\left (\frac {2 \,\mathrm {log}\left (x \right )-1}{\mathrm {log}\left (x \right )}\right )-2 \,\mathrm {log}\left (x \right )+4 x} \] Input:
int(((4*x*log(1/x)^2+2*x*log(1/x))*log((2*log(1/x)+1)/log(1/x))^2+((-8*x*l og(1/x)^2-4*x*log(1/x))*log(x)+(16*x^2+2)*log(1/x)^2+(8*x^2+1)*log(1/x))*l og((2*log(1/x)+1)/log(1/x))+(4*x*log(1/x)^2+2*x*log(1/x))*log(x)^2+((-16*x ^2-2)*log(1/x)^2+(-8*x^2-1)*log(1/x))*log(x)+(16*x^3+2)*log(1/x)^2+(8*x^3+ 1)*log(1/x)-1)/((2*log(1/x)^2+log(1/x))*log((2*log(1/x)+1)/log(1/x))^2+((- 4*log(1/x)^2-2*log(1/x))*log(x)+8*x*log(1/x)^2+4*x*log(1/x))*log((2*log(1/ x)+1)/log(1/x))+(2*log(1/x)^2+log(1/x))*log(x)^2+(-8*x*log(1/x)^2-4*x*log( 1/x))*log(x)+8*x^2*log(1/x)^2+4*x^2*log(1/x)),x)
Output:
(2*log((2*log(x) - 1)/log(x))*x**2 - log((2*log(x) - 1)/log(x)) - 2*log(x) *x**2 + log(x) + 4*x**3)/(2*(log((2*log(x) - 1)/log(x)) - log(x) + 2*x))