\(\int \frac {2 x^2+4 x^3+(2 x+2 x^2+2 x^3) \log (1+x+x^2)+(e^3 (-1-x-x^2)+(-2 x-2 x^2-2 x^3) \log (1+x+x^2)) \log (e^3+2 x \log (1+x+x^2))+(e^3 (-1-x-x^2)+(-2 x-2 x^2-2 x^3) \log (1+x+x^2)) \log (e^3+2 x \log (1+x+x^2)) \log (\log (e^3+2 x \log (1+x+x^2)))}{(e^3 (x^2+x^3+x^4)+(2 x^3+2 x^4+2 x^5) \log (1+x+x^2)) \log (e^3+2 x \log (1+x+x^2))} \, dx\) [1001]

Optimal result

Integrand size = 214, antiderivative size = 22 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {1+\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x} \] Output:

(ln(ln(2*x*ln(x^2+x+1)+exp(3)))+1)/x
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {1}{x}+\frac {\log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{x} \] Input:

Integrate[(2*x^2 + 4*x^3 + (2*x + 2*x^2 + 2*x^3)*Log[1 + x + x^2] + (E^3*( 
-1 - x - x^2) + (-2*x - 2*x^2 - 2*x^3)*Log[1 + x + x^2])*Log[E^3 + 2*x*Log 
[1 + x + x^2]] + (E^3*(-1 - x - x^2) + (-2*x - 2*x^2 - 2*x^3)*Log[1 + x + 
x^2])*Log[E^3 + 2*x*Log[1 + x + x^2]]*Log[Log[E^3 + 2*x*Log[1 + x + x^2]]] 
)/((E^3*(x^2 + x^3 + x^4) + (2*x^3 + 2*x^4 + 2*x^5)*Log[1 + x + x^2])*Log[ 
E^3 + 2*x*Log[1 + x + x^2]]),x]
 

Output:

x^(-1) + Log[Log[E^3 + 2*x*Log[1 + x + x^2]]]/x
 

Rubi [F]

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.

\(\displaystyle \int \frac {4 x^3+2 x^2+\left (2 x^3+2 x^2+2 x\right ) \log \left (x^2+x+1\right )+\left (e^3 \left (-x^2-x-1\right )+\left (-2 x^3-2 x^2-2 x\right ) \log \left (x^2+x+1\right )\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+\left (e^3 \left (-x^2-x-1\right )+\left (-2 x^3-2 x^2-2 x\right ) \log \left (x^2+x+1\right )\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{\left (e^3 \left (x^4+x^3+x^2\right )+\left (2 x^5+2 x^4+2 x^3\right ) \log \left (x^2+x+1\right )\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {4 x^3+2 x^2+\left (2 x^3+2 x^2+2 x\right ) \log \left (x^2+x+1\right )+\left (e^3 \left (-x^2-x-1\right )+\left (-2 x^3-2 x^2-2 x\right ) \log \left (x^2+x+1\right )\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+\left (e^3 \left (-x^2-x-1\right )+\left (-2 x^3-2 x^2-2 x\right ) \log \left (x^2+x+1\right )\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 (2 x+1) x^2-2 \left (x^2+x+1\right ) x \log \left (x^2+x+1\right ) \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )-1\right )-e^3 \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \left (\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )+1\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {4 x^3+2 x^2+2 x^2 \log \left (x^2+x+1\right )-2 x^2 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x^2 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x \log \left (x^2+x+1\right )-2 x \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 x \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )-e^3 \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )+2 x^3 \log \left (x^2+x+1\right )-2 x^3 \log \left (x^2+x+1\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}{x^2 \left (x^2+x+1\right ) \left (2 x \log \left (x^2+x+1\right )+e^3\right ) \log \left (2 x \log \left (x^2+x+1\right )+e^3\right )}-\frac {\log \left (\log \left (2 x \log \left (x^2+x+1\right )+e^3\right )\right )}{x^2}\right )dx\)

Input:

Int[(2*x^2 + 4*x^3 + (2*x + 2*x^2 + 2*x^3)*Log[1 + x + x^2] + (E^3*(-1 - x 
 - x^2) + (-2*x - 2*x^2 - 2*x^3)*Log[1 + x + x^2])*Log[E^3 + 2*x*Log[1 + x 
 + x^2]] + (E^3*(-1 - x - x^2) + (-2*x - 2*x^2 - 2*x^3)*Log[1 + x + x^2])* 
Log[E^3 + 2*x*Log[1 + x + x^2]]*Log[Log[E^3 + 2*x*Log[1 + x + x^2]]])/((E^ 
3*(x^2 + x^3 + x^4) + (2*x^3 + 2*x^4 + 2*x^5)*Log[1 + x + x^2])*Log[E^3 + 
2*x*Log[1 + x + x^2]]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 220.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {\ln \left (\ln \left (2 x \ln \left (x^{2}+x +1\right )+{\mathrm e}^{3}\right )\right )}{x}+\frac {1}{x}\)	\(24\)
parallelrisch	\(\frac {4+4 \ln \left (\ln \left (2 x \ln \left (x^{2}+x +1\right )+{\mathrm e}^{3}\right )\right )}{4 x}\)	\(25\)

Input:

int((((-2*x^3-2*x^2-2*x)*ln(x^2+x+1)+(-x^2-x-1)*exp(3))*ln(2*x*ln(x^2+x+1) 
+exp(3))*ln(ln(2*x*ln(x^2+x+1)+exp(3)))+((-2*x^3-2*x^2-2*x)*ln(x^2+x+1)+(- 
x^2-x-1)*exp(3))*ln(2*x*ln(x^2+x+1)+exp(3))+(2*x^3+2*x^2+2*x)*ln(x^2+x+1)+ 
4*x^3+2*x^2)/((2*x^5+2*x^4+2*x^3)*ln(x^2+x+1)+(x^4+x^3+x^2)*exp(3))/ln(2*x 
*ln(x^2+x+1)+exp(3)),x,method=_RETURNVERBOSE)
 

Output:

1/x*ln(ln(2*x*ln(x^2+x+1)+exp(3)))+1/x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {\log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 1}{x} \] Input:

integrate((((-2*x^3-2*x^2-2*x)*log(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log 
(x^2+x+1)+exp(3))*log(log(2*x*log(x^2+x+1)+exp(3)))+((-2*x^3-2*x^2-2*x)*lo 
g(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log(x^2+x+1)+exp(3))+(2*x^3+2*x^2+2* 
x)*log(x^2+x+1)+4*x^3+2*x^2)/((2*x^5+2*x^4+2*x^3)*log(x^2+x+1)+(x^4+x^3+x^ 
2)*exp(3))/log(2*x*log(x^2+x+1)+exp(3)),x, algorithm="fricas")
 

Output:

(log(log(2*x*log(x^2 + x + 1) + e^3)) + 1)/x
 

Sympy [A] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {\log {\left (\log {\left (2 x \log {\left (x^{2} + x + 1 \right )} + e^{3} \right )} \right )}}{x} + \frac {1}{x} \] Input:

integrate((((-2*x**3-2*x**2-2*x)*ln(x**2+x+1)+(-x**2-x-1)*exp(3))*ln(2*x*l 
n(x**2+x+1)+exp(3))*ln(ln(2*x*ln(x**2+x+1)+exp(3)))+((-2*x**3-2*x**2-2*x)* 
ln(x**2+x+1)+(-x**2-x-1)*exp(3))*ln(2*x*ln(x**2+x+1)+exp(3))+(2*x**3+2*x** 
2+2*x)*ln(x**2+x+1)+4*x**3+2*x**2)/((2*x**5+2*x**4+2*x**3)*ln(x**2+x+1)+(x 
**4+x**3+x**2)*exp(3))/ln(2*x*ln(x**2+x+1)+exp(3)),x)
 

Output:

log(log(2*x*log(x**2 + x + 1) + exp(3)))/x + 1/x
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {\log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 1}{x} \] Input:

integrate((((-2*x^3-2*x^2-2*x)*log(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log 
(x^2+x+1)+exp(3))*log(log(2*x*log(x^2+x+1)+exp(3)))+((-2*x^3-2*x^2-2*x)*lo 
g(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log(x^2+x+1)+exp(3))+(2*x^3+2*x^2+2* 
x)*log(x^2+x+1)+4*x^3+2*x^2)/((2*x^5+2*x^4+2*x^3)*log(x^2+x+1)+(x^4+x^3+x^ 
2)*exp(3))/log(2*x*log(x^2+x+1)+exp(3)),x, algorithm="maxima")
 

Output:

(log(log(2*x*log(x^2 + x + 1) + e^3)) + 1)/x
 

Giac [F]

\[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\int { \frac {4 \, x^{3} - {\left ({\left (x^{2} + x + 1\right )} e^{3} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right ) \log \left (\log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )\right ) + 2 \, x^{2} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right ) - {\left ({\left (x^{2} + x + 1\right )} e^{3} + 2 \, {\left (x^{3} + x^{2} + x\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )}{{\left ({\left (x^{4} + x^{3} + x^{2}\right )} e^{3} + 2 \, {\left (x^{5} + x^{4} + x^{3}\right )} \log \left (x^{2} + x + 1\right )\right )} \log \left (2 \, x \log \left (x^{2} + x + 1\right ) + e^{3}\right )} \,d x } \] Input:

integrate((((-2*x^3-2*x^2-2*x)*log(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log 
(x^2+x+1)+exp(3))*log(log(2*x*log(x^2+x+1)+exp(3)))+((-2*x^3-2*x^2-2*x)*lo 
g(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log(x^2+x+1)+exp(3))+(2*x^3+2*x^2+2* 
x)*log(x^2+x+1)+4*x^3+2*x^2)/((2*x^5+2*x^4+2*x^3)*log(x^2+x+1)+(x^4+x^3+x^ 
2)*exp(3))/log(2*x*log(x^2+x+1)+exp(3)),x, algorithm="giac")
 

Output:

integrate((4*x^3 - ((x^2 + x + 1)*e^3 + 2*(x^3 + x^2 + x)*log(x^2 + x + 1) 
)*log(2*x*log(x^2 + x + 1) + e^3)*log(log(2*x*log(x^2 + x + 1) + e^3)) + 2 
*x^2 + 2*(x^3 + x^2 + x)*log(x^2 + x + 1) - ((x^2 + x + 1)*e^3 + 2*(x^3 + 
x^2 + x)*log(x^2 + x + 1))*log(2*x*log(x^2 + x + 1) + e^3))/(((x^4 + x^3 + 
 x^2)*e^3 + 2*(x^5 + x^4 + x^3)*log(x^2 + x + 1))*log(2*x*log(x^2 + x + 1) 
 + e^3)), x)
 

Mupad [B] (verification not implemented)

Time = 2.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {\ln \left (\ln \left ({\mathrm {e}}^3+2\,x\,\ln \left (x^2+x+1\right )\right )\right )+1}{x} \] Input:

int((log(x + x^2 + 1)*(2*x + 2*x^2 + 2*x^3) - log(exp(3) + 2*x*log(x + x^2 
 + 1))*(log(x + x^2 + 1)*(2*x + 2*x^2 + 2*x^3) + exp(3)*(x + x^2 + 1)) + 2 
*x^2 + 4*x^3 - log(exp(3) + 2*x*log(x + x^2 + 1))*log(log(exp(3) + 2*x*log 
(x + x^2 + 1)))*(log(x + x^2 + 1)*(2*x + 2*x^2 + 2*x^3) + exp(3)*(x + x^2 
+ 1)))/(log(exp(3) + 2*x*log(x + x^2 + 1))*(log(x + x^2 + 1)*(2*x^3 + 2*x^ 
4 + 2*x^5) + exp(3)*(x^2 + x^3 + x^4))),x)
 

Output:

(log(log(exp(3) + 2*x*log(x + x^2 + 1))) + 1)/x
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2+4 x^3+\left (2 x+2 x^2+2 x^3\right ) \log \left (1+x+x^2\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )+\left (e^3 \left (-1-x-x^2\right )+\left (-2 x-2 x^2-2 x^3\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right ) \log \left (\log \left (e^3+2 x \log \left (1+x+x^2\right )\right )\right )}{\left (e^3 \left (x^2+x^3+x^4\right )+\left (2 x^3+2 x^4+2 x^5\right ) \log \left (1+x+x^2\right )\right ) \log \left (e^3+2 x \log \left (1+x+x^2\right )\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (2 \,\mathrm {log}\left (x^{2}+x +1\right ) x +e^{3}\right )\right )+1}{x} \] Input:

int((((-2*x^3-2*x^2-2*x)*log(x^2+x+1)+(-x^2-x-1)*exp(3))*log(2*x*log(x^2+x 
+1)+exp(3))*log(log(2*x*log(x^2+x+1)+exp(3)))+((-2*x^3-2*x^2-2*x)*log(x^2+ 
x+1)+(-x^2-x-1)*exp(3))*log(2*x*log(x^2+x+1)+exp(3))+(2*x^3+2*x^2+2*x)*log 
(x^2+x+1)+4*x^3+2*x^2)/((2*x^5+2*x^4+2*x^3)*log(x^2+x+1)+(x^4+x^3+x^2)*exp 
(3))/log(2*x*log(x^2+x+1)+exp(3)),x)
 

Output:

(log(log(2*log(x**2 + x + 1)*x + e**3)) + 1)/x