\(\int \frac {260 x+62 x^2+2 x^3+(e^x (125 x+30 x^2+x^3)+e^x (125+30 x+x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(125 x+30 x^2+x^3+(125+30 x+x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))+((e^x (-25 x-x^2)+e^x (-25-x) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(-25 x^2-x^3+(-25 x-x^2) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))) \log (e^x+x \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x))))}{(e^x (625 x+275 x^2+35 x^3+x^4)+e^x (625+275 x+35 x^2+x^3) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x))+(625 x^2+275 x^3+35 x^4+x^5+(625 x+275 x^2+35 x^3+x^4) \log (25+x)) \log (x^2+2 x \log (25+x)+\log ^2(25+x)) \log (\log (x^2+2 x \log (25+x)+\log ^2(25+x)))} \, dx\) [283]

Optimal result

Integrand size = 411, antiderivative size = 23 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{5+x} \] Output:

ln(x*ln(ln((ln(x+25)+x)^2))+exp(x))/(5+x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (e^x+x \log \left (\log \left ((x+\log (25+x))^2\right )\right )\right )}{5+x} \] Input:

Integrate[(260*x + 62*x^2 + 2*x^3 + (E^x*(125*x + 30*x^2 + x^3) + E^x*(125 
 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + ( 
125*x + 30*x^2 + x^3 + (125 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log[2 
5 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]] + 
((E^x*(-25*x - x^2) + E^x*(-25 - x)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] 
 + Log[25 + x]^2] + (-25*x^2 - x^3 + (-25*x - x^2)*Log[25 + x])*Log[x^2 + 
2*x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + 
x]^2]])*Log[E^x + x*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]]])/((E^ 
x*(625*x + 275*x^2 + 35*x^3 + x^4) + E^x*(625 + 275*x + 35*x^2 + x^3)*Log[ 
25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (625*x^2 + 275*x^3 + 
 35*x^4 + x^5 + (625*x + 275*x^2 + 35*x^3 + x^4)*Log[25 + x])*Log[x^2 + 2* 
x*Log[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x] 
^2]]),x]
 

Output:

Log[E^x + x*Log[Log[(x + Log[25 + x])^2]]]/(5 + 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.

\(\displaystyle \int \frac {2 x^3+62 x^2+\left (e^x \left (x^2+30 x+125\right ) \log (x+25)+e^x \left (x^3+30 x^2+125 x\right )\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )+\left (x^3+30 x^2+\left (x^2+30 x+125\right ) \log (x+25)+125 x\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right ) \log \left (\log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )\right )+\left (\left (e^x \left (-x^2-25 x\right )+e^x (-x-25) \log (x+25)\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )+\left (-x^3-25 x^2+\left (-x^2-25 x\right ) \log (x+25)\right ) \log \left (\log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )\right ) \log \left (x \log \left (\log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )\right )+e^x\right )+260 x}{\left (e^x \left (x^3+35 x^2+275 x+625\right ) \log (x+25)+e^x \left (x^4+35 x^3+275 x^2+625 x\right )\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )+\left (x^5+35 x^4+275 x^3+625 x^2+\left (x^4+35 x^3+275 x^2+625 x\right ) \log (x+25)\right ) \log \left (\log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )\right ) \log \left (x^2+\log ^2(x+25)+2 x \log (x+25)\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x^3+62 x^2+e^x \left (x^2+30 x+125\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right )+\left (x^2+30 x+125\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+260 x-(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right ) \log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}{(x+5)^2 (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\frac {(x+5) \left (\left (x^2+24 x-25\right ) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x (x+26)\right )}{(x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}+x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x-\log \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )+5}{(x+5)^2}-\frac {x^3 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-2 x^2+x^2 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )+24 x^2 \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-52 x+24 x \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 x \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )-25 \log (x+25) \log \left ((x+\log (x+25))^2\right ) \log \left (\log \left ((x+\log (x+25))^2\right )\right )}{(x+5) (x+25) (x+\log (x+25)) \log \left ((x+\log (x+25))^2\right ) \left (e^x+x \log \left (\log \left ((x+\log (x+25))^2\right )\right )\right )}\right )dx\)

Input:

Int[(260*x + 62*x^2 + 2*x^3 + (E^x*(125*x + 30*x^2 + x^3) + E^x*(125 + 30* 
x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (125*x 
+ 30*x^2 + x^3 + (125 + 30*x + x^2)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] 
 + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]] + ((E^x* 
(-25*x - x^2) + E^x*(-25 - x)*Log[25 + x])*Log[x^2 + 2*x*Log[25 + x] + Log 
[25 + x]^2] + (-25*x^2 - x^3 + (-25*x - x^2)*Log[25 + x])*Log[x^2 + 2*x*Lo 
g[25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]] 
)*Log[E^x + x*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]]])/((E^x*(625 
*x + 275*x^2 + 35*x^3 + x^4) + E^x*(625 + 275*x + 35*x^2 + x^3)*Log[25 + x 
])*Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2] + (625*x^2 + 275*x^3 + 35*x^ 
4 + x^5 + (625*x + 275*x^2 + 35*x^3 + x^4)*Log[25 + x])*Log[x^2 + 2*x*Log[ 
25 + x] + Log[25 + x]^2]*Log[Log[x^2 + 2*x*Log[25 + x] + Log[25 + x]^2]]), 
x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91

\[\frac {\ln \left (x \ln \left (2 \ln \left (\ln \left (x +25\right )+x \right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )^{2}\right )+\operatorname {csgn}\left (i \left (\ln \left (x +25\right )+x \right )\right )\right )}^{2}}{2}\right )+{\mathrm e}^{x}\right )}{5+x}\]

Input:

int(((((-x^2-25*x)*ln(x+25)-x^3-25*x^2)*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)*ln 
(ln(ln(x+25)^2+2*x*ln(x+25)+x^2))+((-x-25)*exp(x)*ln(x+25)+(-x^2-25*x)*exp 
(x))*ln(ln(x+25)^2+2*x*ln(x+25)+x^2))*ln(x*ln(ln(ln(x+25)^2+2*x*ln(x+25)+x 
^2))+exp(x))+((x^2+30*x+125)*ln(x+25)+x^3+30*x^2+125*x)*ln(ln(x+25)^2+2*x* 
ln(x+25)+x^2)*ln(ln(ln(x+25)^2+2*x*ln(x+25)+x^2))+((x^2+30*x+125)*exp(x)*l 
n(x+25)+(x^3+30*x^2+125*x)*exp(x))*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)+2*x^3+6 
2*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*ln(x+25)+x^5+35*x^4+275*x^3+625* 
x^2)*ln(ln(x+25)^2+2*x*ln(x+25)+x^2)*ln(ln(ln(x+25)^2+2*x*ln(x+25)+x^2))+( 
(x^3+35*x^2+275*x+625)*exp(x)*ln(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))* 
ln(ln(x+25)^2+2*x*ln(x+25)+x^2)),x)
 

Output:

1/(5+x)*ln(x*ln(2*ln(ln(x+25)+x)-1/2*I*Pi*csgn(I*(ln(x+25)+x)^2)*(-csgn(I* 
(ln(x+25)+x)^2)+csgn(I*(ln(x+25)+x)))^2)+exp(x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x \log \left (\log \left (x^{2} + 2 \, x \log \left (x + 25\right ) + \log \left (x + 25\right )^{2}\right )\right ) + e^{x}\right )}{x + 5} \] Input:

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+2 
5)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+ 
(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(log( 
x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+1 
25*x)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25) 
+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log(log 
(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625* 
x)*log(x+25)+x^5+35*x^4+275*x^3+625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2 
)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*exp(x)*l 
og(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+ 
x^2)),x, algorithm="fricas")
 

Output:

log(x*log(log(x^2 + 2*x*log(x + 25) + log(x + 25)^2)) + e^x)/(x + 5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((((-x**2-25*x)*ln(x+25)-x**3-25*x**2)*ln(ln(x+25)**2+2*x*ln(x+2 
5)+x**2)*ln(ln(ln(x+25)**2+2*x*ln(x+25)+x**2))+((-x-25)*exp(x)*ln(x+25)+(- 
x**2-25*x)*exp(x))*ln(ln(x+25)**2+2*x*ln(x+25)+x**2))*ln(x*ln(ln(ln(x+25)* 
*2+2*x*ln(x+25)+x**2))+exp(x))+((x**2+30*x+125)*ln(x+25)+x**3+30*x**2+125* 
x)*ln(ln(x+25)**2+2*x*ln(x+25)+x**2)*ln(ln(ln(x+25)**2+2*x*ln(x+25)+x**2)) 
+((x**2+30*x+125)*exp(x)*ln(x+25)+(x**3+30*x**2+125*x)*exp(x))*ln(ln(x+25) 
**2+2*x*ln(x+25)+x**2)+2*x**3+62*x**2+260*x)/(((x**4+35*x**3+275*x**2+625* 
x)*ln(x+25)+x**5+35*x**4+275*x**3+625*x**2)*ln(ln(x+25)**2+2*x*ln(x+25)+x* 
*2)*ln(ln(ln(x+25)**2+2*x*ln(x+25)+x**2))+((x**3+35*x**2+275*x+625)*exp(x) 
*ln(x+25)+(x**4+35*x**3+275*x**2+625*x)*exp(x))*ln(ln(x+25)**2+2*x*ln(x+25 
)+x**2)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x {\left (\log \left (2\right ) + \log \left (\log \left (x + \log \left (x + 25\right )\right )\right )\right )} + e^{x}\right )}{x + 5} \] Input:

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+2 
5)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+ 
(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(log( 
x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+1 
25*x)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25) 
+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log(log 
(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625* 
x)*log(x+25)+x^5+35*x^4+275*x^3+625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2 
)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*exp(x)*l 
og(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+ 
x^2)),x, algorithm="maxima")
 

Output:

log(x*(log(2) + log(log(x + log(x + 25)))) + e^x)/(x + 5)
 

Giac [A] (verification not implemented)

Time = 21.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\log \left (x \log \left (\log \left (x^{2} + 2 \, x \log \left (x + 25\right ) + \log \left (x + 25\right )^{2}\right )\right ) + e^{x}\right )}{x + 5} \] Input:

integrate(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+2 
5)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+ 
(-x^2-25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(log( 
x+25)^2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+1 
25*x)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25) 
+x^2))+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log(log 
(x+25)^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625* 
x)*log(x+25)+x^5+35*x^4+275*x^3+625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2 
)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*exp(x)*l 
og(x+25)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+ 
x^2)),x, algorithm="giac")
 

Output:

log(x*log(log(x^2 + 2*x*log(x + 25) + log(x + 25)^2)) + e^x)/(x + 5)
 

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\ln \left ({\mathrm {e}}^x+x\,\ln \left (\ln \left (x^2+2\,x\,\ln \left (x+25\right )+{\ln \left (x+25\right )}^2\right )\right )\right )\,\left (x^2+25\,x\right )\,\left (x^2+30\,x+125\right )}{x\,{\left (x+5\right )}^2\,{\left (x+25\right )}^2} \] Input:

int((260*x + log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*(exp(x)*(125*x + 3 
0*x^2 + x^3) + log(x + 25)*exp(x)*(30*x + x^2 + 125)) - log(exp(x) + x*log 
(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)))*(log(2*x*log(x + 25) + log(x 
 + 25)^2 + x^2)*(exp(x)*(25*x + x^2) + log(x + 25)*exp(x)*(x + 25)) + log( 
2*x*log(x + 25) + log(x + 25)^2 + x^2)*log(log(2*x*log(x + 25) + log(x + 2 
5)^2 + x^2))*(log(x + 25)*(25*x + x^2) + 25*x^2 + x^3)) + 62*x^2 + 2*x^3 + 
 log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*log(log(2*x*log(x + 25) + log( 
x + 25)^2 + x^2))*(125*x + log(x + 25)*(30*x + x^2 + 125) + 30*x^2 + x^3)) 
/(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)*(exp(x)*(625*x + 275*x^2 + 35 
*x^3 + x^4) + log(x + 25)*exp(x)*(275*x + 35*x^2 + x^3 + 625)) + log(2*x*l 
og(x + 25) + log(x + 25)^2 + x^2)*log(log(2*x*log(x + 25) + log(x + 25)^2 
+ x^2))*(log(x + 25)*(625*x + 275*x^2 + 35*x^3 + x^4) + 625*x^2 + 275*x^3 
+ 35*x^4 + x^5)),x)
                                                                                    
                                                                                    
 

Output:

(log(exp(x) + x*log(log(2*x*log(x + 25) + log(x + 25)^2 + x^2)))*(25*x + x 
^2)*(30*x + x^2 + 125))/(x*(x + 5)^2*(x + 25)^2)
 

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {260 x+62 x^2+2 x^3+\left (e^x \left (125 x+30 x^2+x^3\right )+e^x \left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (125 x+30 x^2+x^3+\left (125+30 x+x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )+\left (\left (e^x \left (-25 x-x^2\right )+e^x (-25-x) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (-25 x^2-x^3+\left (-25 x-x^2\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right ) \log \left (e^x+x \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )\right )}{\left (e^x \left (625 x+275 x^2+35 x^3+x^4\right )+e^x \left (625+275 x+35 x^2+x^3\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )+\left (625 x^2+275 x^3+35 x^4+x^5+\left (625 x+275 x^2+35 x^3+x^4\right ) \log (25+x)\right ) \log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right ) \log \left (\log \left (x^2+2 x \log (25+x)+\log ^2(25+x)\right )\right )} \, dx=\frac {\mathrm {log}\left (e^{x}+\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x +25\right )^{2}+2 \,\mathrm {log}\left (x +25\right ) x +x^{2}\right )\right ) x \right )}{x +5} \] Input:

int(((((-x^2-25*x)*log(x+25)-x^3-25*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2 
)*log(log(log(x+25)^2+2*x*log(x+25)+x^2))+((-x-25)*exp(x)*log(x+25)+(-x^2- 
25*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2))*log(x*log(log(log(x+25)^ 
2+2*x*log(x+25)+x^2))+exp(x))+((x^2+30*x+125)*log(x+25)+x^3+30*x^2+125*x)* 
log(log(x+25)^2+2*x*log(x+25)+x^2)*log(log(log(x+25)^2+2*x*log(x+25)+x^2)) 
+((x^2+30*x+125)*exp(x)*log(x+25)+(x^3+30*x^2+125*x)*exp(x))*log(log(x+25) 
^2+2*x*log(x+25)+x^2)+2*x^3+62*x^2+260*x)/(((x^4+35*x^3+275*x^2+625*x)*log 
(x+25)+x^5+35*x^4+275*x^3+625*x^2)*log(log(x+25)^2+2*x*log(x+25)+x^2)*log( 
log(log(x+25)^2+2*x*log(x+25)+x^2))+((x^3+35*x^2+275*x+625)*exp(x)*log(x+2 
5)+(x^4+35*x^3+275*x^2+625*x)*exp(x))*log(log(x+25)^2+2*x*log(x+25)+x^2)), 
x)
 

Output:

log(e**x + log(log(log(x + 25)**2 + 2*log(x + 25)*x + x**2))*x)/(x + 5)