\(\int \frac {2 x+4 x^2+2 x^3+(2 x+6 x^2+4 x^3) \log (x) \log (\log (x)) \log (\log ^2(\log (x)))}{-e^{25} \log (x) \log (\log (x))+(x^2+2 x^3+x^4) \log (x) \log (\log (x)) \log (\log ^2(\log (x)))} \, dx\) [2670]

Optimal result

Integrand size = 78, antiderivative size = 20 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (e^{25}-\left (x+x^2\right )^2 \log \left (\log ^2(\log (x))\right )\right ) \] Output:

ln(exp(25)-(x^2+x)^2*ln(ln(ln(x))^2))
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (e^{25}-x^2 \log \left (\log ^2(\log (x))\right )-2 x^3 \log \left (\log ^2(\log (x))\right )-x^4 \log \left (\log ^2(\log (x))\right )\right ) \] Input:

Integrate[(2*x + 4*x^2 + 2*x^3 + (2*x + 6*x^2 + 4*x^3)*Log[x]*Log[Log[x]]* 
Log[Log[Log[x]]^2])/(-(E^25*Log[x]*Log[Log[x]]) + (x^2 + 2*x^3 + x^4)*Log[ 
x]*Log[Log[x]]*Log[Log[Log[x]]^2]),x]
 

Output:

Log[E^25 - x^2*Log[Log[Log[x]]^2] - 2*x^3*Log[Log[Log[x]]^2] - x^4*Log[Log 
[Log[x]]^2]]
 

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.

\(\displaystyle \int \frac {2 x^3+4 x^2+\left (4 x^3+6 x^2+2 x\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+2 x}{\left (x^4+2 x^3+x^2\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )-e^{25} \log (x) \log (\log (x))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {2 x (x+1) \left (-x-2 x \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )-\log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )-1\right )}{e^{25} \log (x) \log (\log (x))-\left (x^4+2 x^3+x^2\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {x (x+1) \left (2 \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right ) x+x+\log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{e^{25} \log (x) \log (\log (x))-\left (x^4+2 x^3+x^2\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {x (x+1) \left (2 \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right ) x+x+\log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{e^{25} \log (x) \log (\log (x))-\left (x^4+2 x^3+x^2\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {x (x+1) \left (x+(2 x+1) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )+1\right )}{\log (x) \log (\log (x)) \left (e^{25}-x^2 (x+1)^2 \log \left (\log ^2(\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {-2 x-1}{x (x+1)}+\frac {-x^5-3 x^4-3 x^3-x^2-2 e^{25} \log (x) \log (\log (x)) x-e^{25} \log (x) \log (\log (x))}{x (x+1) \log (x) \log (\log (x)) \left (\log \left (\log ^2(\log (x))\right ) x^4+2 \log \left (\log ^2(\log (x))\right ) x^3+\log \left (\log ^2(\log (x))\right ) x^2-e^{25}\right )}\right )dx\)

Input:

Int[(2*x + 4*x^2 + 2*x^3 + (2*x + 6*x^2 + 4*x^3)*Log[x]*Log[Log[x]]*Log[Lo 
g[Log[x]]^2])/(-(E^25*Log[x]*Log[Log[x]]) + (x^2 + 2*x^3 + x^4)*Log[x]*Log 
[Log[x]]*Log[Log[Log[x]]^2]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 54.86 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90

method	result	size
parallelrisch	\(\ln \left (\ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right ) x^{4}+2 \ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right ) x^{3}+x^{2} \ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right )-{\mathrm e}^{25}\right )\)	\(38\)
risch	\(2 \ln \left (x^{2}+x \right )+\ln \left (\ln \left (\ln \left (\ln \left (x \right )\right )\right )-\frac {i \left (\pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-2 \pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+\pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}+2 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-4 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+2 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}+\pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-2 \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+\pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}-2 i {\mathrm e}^{25}\right )}{4 x^{2} \left (x^{2}+2 x +1\right )}\right )\)	\(229\)

Input:

int(((4*x^3+6*x^2+2*x)*ln(x)*ln(ln(x))*ln(ln(ln(x))^2)+2*x^3+4*x^2+2*x)/(( 
x^4+2*x^3+x^2)*ln(x)*ln(ln(x))*ln(ln(ln(x))^2)-exp(25)*ln(x)*ln(ln(x))),x, 
method=_RETURNVERBOSE)
 

Output:

ln(ln(ln(ln(x))^2)*x^4+2*ln(ln(ln(x))^2)*x^3+x^2*ln(ln(ln(x))^2)-exp(25))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \, \log \left (x^{2} + x\right ) + \log \left (\frac {{\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - e^{25}}{x^{4} + 2 \, x^{3} + x^{2}}\right ) \] Input:

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4 
*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*log(log(x))*log(log(log(x))^2)-exp(25)*l 
og(x)*log(log(x))),x, algorithm="fricas")
 

Output:

2*log(x^2 + x) + log(((x^4 + 2*x^3 + x^2)*log(log(log(x))^2) - e^25)/(x^4 
+ 2*x^3 + x^2))
 

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \log {\left (x^{2} + x \right )} + \log {\left (\log {\left (\log {\left (\log {\left (x \right )} \right )}^{2} \right )} - \frac {e^{25}}{x^{4} + 2 x^{3} + x^{2}} \right )} \] Input:

integrate(((4*x**3+6*x**2+2*x)*ln(x)*ln(ln(x))*ln(ln(ln(x))**2)+2*x**3+4*x 
**2+2*x)/((x**4+2*x**3+x**2)*ln(x)*ln(ln(x))*ln(ln(ln(x))**2)-exp(25)*ln(x 
)*ln(ln(x))),x)
 

Output:

2*log(x**2 + x) + log(log(log(log(x))**2) - exp(25)/(x**4 + 2*x**3 + x**2) 
)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \, \log \left (x + 1\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - e^{25}}{2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}\right ) \] Input:

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4 
*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*log(log(x))*log(log(log(x))^2)-exp(25)*l 
og(x)*log(log(x))),x, algorithm="maxima")
 

Output:

2*log(x + 1) + 2*log(x) + log(1/2*(2*(x^4 + 2*x^3 + x^2)*log(log(log(x))) 
- e^25)/(x^4 + 2*x^3 + x^2))
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (-x^{4} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - 2 \, x^{3} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - x^{2} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) + e^{25}\right ) \] Input:

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4 
*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*log(log(x))*log(log(log(x))^2)-exp(25)*l 
og(x)*log(log(x))),x, algorithm="giac")
 

Output:

log(-x^4*log(log(log(x))^2) - 2*x^3*log(log(log(x))^2) - x^2*log(log(log(x 
))^2) + e^25)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=-\int \frac {2\,x+4\,x^2+2\,x^3+\ln \left ({\ln \left (\ln \left (x\right )\right )}^2\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (4\,x^3+6\,x^2+2\,x\right )}{\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{25}\,\ln \left (x\right )-\ln \left ({\ln \left (\ln \left (x\right )\right )}^2\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (x^4+2\,x^3+x^2\right )} \,d x \] Input:

int(-(2*x + 4*x^2 + 2*x^3 + log(log(log(x))^2)*log(log(x))*log(x)*(2*x + 6 
*x^2 + 4*x^3))/(log(log(x))*exp(25)*log(x) - log(log(log(x))^2)*log(log(x) 
)*log(x)*(x^2 + 2*x^3 + x^4)),x)
 

Output:

-int((2*x + 4*x^2 + 2*x^3 + log(log(log(x))^2)*log(log(x))*log(x)*(2*x + 6 
*x^2 + 4*x^3))/(log(log(x))*exp(25)*log(x) - log(log(log(x))^2)*log(log(x) 
)*log(x)*(x^2 + 2*x^3 + x^4)), x)
 

Reduce [F]

\[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx =\text {Too large to display} \] Input:

int(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4*x^2+2 
*x)/((x^4+2*x^3+x^2)*log(x)*log(log(x))*log(log(log(x))^2)-exp(25)*log(x)* 
log(log(x))),x)
 

Output:

2*(int(x**3/(log(log(log(x))**2)*log(log(x))*log(x)*x**4 + 2*log(log(log(x 
))**2)*log(log(x))*log(x)*x**3 + log(log(log(x))**2)*log(log(x))*log(x)*x* 
*2 - log(log(x))*log(x)*e**25),x) + 2*int(x**2/(log(log(log(x))**2)*log(lo 
g(x))*log(x)*x**4 + 2*log(log(log(x))**2)*log(log(x))*log(x)*x**3 + log(lo 
g(log(x))**2)*log(log(x))*log(x)*x**2 - log(log(x))*log(x)*e**25),x) + 2*i 
nt((log(log(log(x))**2)*x**3)/(log(log(log(x))**2)*x**4 + 2*log(log(log(x) 
)**2)*x**3 + log(log(log(x))**2)*x**2 - e**25),x) + 3*int((log(log(log(x)) 
**2)*x**2)/(log(log(log(x))**2)*x**4 + 2*log(log(log(x))**2)*x**3 + log(lo 
g(log(x))**2)*x**2 - e**25),x) + int((log(log(log(x))**2)*x)/(log(log(log( 
x))**2)*x**4 + 2*log(log(log(x))**2)*x**3 + log(log(log(x))**2)*x**2 - e** 
25),x) + int(x/(log(log(log(x))**2)*log(log(x))*log(x)*x**4 + 2*log(log(lo 
g(x))**2)*log(log(x))*log(x)*x**3 + log(log(log(x))**2)*log(log(x))*log(x) 
*x**2 - log(log(x))*log(x)*e**25),x))