\(\int \frac {24 x+48 x^2+8 x^3+e^x (-4 x-44 x^2-28 x^3-4 x^4)+e^{2 x} (-4 x+10 x^2+12 x^3+2 x^4)+(-40-68 x-12 x^2+e^{2 x} (-15 x-13 x^2-2 x^3)+e^x (20+64 x+32 x^2+4 x^3)) \log (5+x)+\log ^2(x^2) (-4 x+(-5 x-x^2) \log (5+x))+\log (x^2) (4 x+24 x^2+4 x^3+e^x (8 x-10 x^2-12 x^3-2 x^4)+(-20-44 x-8 x^2+e^x (20 x+14 x^2+2 x^3)) \log (5+x))}{5 x^3+x^4+(-10 x^2-2 x^3) \log (5+x)+(5 x+x^2) \log ^2(5+x)} \, dx\) [11]

Optimal result

Integrand size = 258, antiderivative size = 29 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=2+\frac {(1+x) \left (-2+e^x-\log \left (x^2\right )\right )^2}{x-\log (5+x)} \] Output:

2+(exp(x)-2-ln(x^2))^2/(x-ln(5+x))*(1+x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=-\frac {(1+x) \left (-2+e^x-\log \left (x^2\right )\right )^2}{-x+\log (5+x)} \] Input:

Integrate[(24*x + 48*x^2 + 8*x^3 + E^x*(-4*x - 44*x^2 - 28*x^3 - 4*x^4) + 
E^(2*x)*(-4*x + 10*x^2 + 12*x^3 + 2*x^4) + (-40 - 68*x - 12*x^2 + E^(2*x)* 
(-15*x - 13*x^2 - 2*x^3) + E^x*(20 + 64*x + 32*x^2 + 4*x^3))*Log[5 + x] + 
Log[x^2]^2*(-4*x + (-5*x - x^2)*Log[5 + x]) + Log[x^2]*(4*x + 24*x^2 + 4*x 
^3 + E^x*(8*x - 10*x^2 - 12*x^3 - 2*x^4) + (-20 - 44*x - 8*x^2 + E^x*(20*x 
 + 14*x^2 + 2*x^3))*Log[5 + x]))/(5*x^3 + x^4 + (-10*x^2 - 2*x^3)*Log[5 + 
x] + (5*x + x^2)*Log[5 + x]^2),x]
 

Output:

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

\(\displaystyle \int \frac {8 x^3+48 x^2+\log ^2\left (x^2\right ) \left (\left (-x^2-5 x\right ) \log (x+5)-4 x\right )+\left (-12 x^2+e^{2 x} \left (-2 x^3-13 x^2-15 x\right )+e^x \left (4 x^3+32 x^2+64 x+20\right )-68 x-40\right ) \log (x+5)+e^x \left (-4 x^4-28 x^3-44 x^2-4 x\right )+e^{2 x} \left (2 x^4+12 x^3+10 x^2-4 x\right )+\log \left (x^2\right ) \left (4 x^3+24 x^2+\left (-8 x^2+e^x \left (2 x^3+14 x^2+20 x\right )-44 x-20\right ) \log (x+5)+e^x \left (-2 x^4-12 x^3-10 x^2+8 x\right )+4 x\right )+24 x}{x^4+5 x^3+\left (x^2+5 x\right ) \log ^2(x+5)+\left (-2 x^3-10 x^2\right ) \log (x+5)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log \left (x^2\right )-e^x+2\right ) \left ((x+5) \left (2 e^x x^2+3 \left (e^x-2\right ) x-4\right ) \log (x+5)-x \log \left (x^2\right ) ((x+5) \log (x+5)+4)-2 x \left (e^x \left (x^3+6 x^2+5 x-2\right )-2 \left (x^2+6 x+3\right )\right )\right )}{x (x+5) (x-\log (x+5))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x \log (x+5)+5 \log (x+5)+4) \log ^2\left (x^2\right )}{(x+5) (x-\log (x+5))^2}-\frac {4 \log (x+5) \log \left (x^2\right )}{x (x-\log (x+5))^2}-\frac {6 \log (x+5) \log \left (x^2\right )}{(x-\log (x+5))^2}-\frac {2 (x \log (x+5)+5 \log (x+5)+4) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {4 \left (x^2+6 x+3\right ) \log \left (x^2\right )}{(x+5) (x-\log (x+5))^2}+\frac {8 \left (x^2+6 x+3\right )}{(x+5) (x-\log (x+5))^2}+\frac {e^{2 x} \left (2 x^3+12 x^2-2 x^2 \log (x+5)+10 x-13 x \log (x+5)-15 \log (x+5)-4\right )}{(x+5) (x-\log (x+5))^2}-\frac {2 e^x \left (2 x^4+14 x^3-2 x^3 \log (x+5)+22 x^2+5 x^2 \log \left (x^2\right )-7 x^2 \log \left (x^2\right ) \log (x+5)-16 x^2 \log (x+5)-4 x \log \left (x^2\right )-10 x \log \left (x^2\right ) \log (x+5)+x^4 \log \left (x^2\right )+6 x^3 \log \left (x^2\right )-x^3 \log \left (x^2\right ) \log (x+5)+2 x-32 x \log (x+5)-10 \log (x+5)\right )}{x (x+5) (x-\log (x+5))^2}-\frac {8 \log (x+5)}{x (x-\log (x+5))^2}-\frac {12 \log (x+5)}{(x-\log (x+5))^2}\right )dx\)

Input:

Int[(24*x + 48*x^2 + 8*x^3 + E^x*(-4*x - 44*x^2 - 28*x^3 - 4*x^4) + E^(2*x 
)*(-4*x + 10*x^2 + 12*x^3 + 2*x^4) + (-40 - 68*x - 12*x^2 + E^(2*x)*(-15*x 
 - 13*x^2 - 2*x^3) + E^x*(20 + 64*x + 32*x^2 + 4*x^3))*Log[5 + x] + Log[x^ 
2]^2*(-4*x + (-5*x - x^2)*Log[5 + x]) + Log[x^2]*(4*x + 24*x^2 + 4*x^3 + E 
^x*(8*x - 10*x^2 - 12*x^3 - 2*x^4) + (-20 - 44*x - 8*x^2 + E^x*(20*x + 14* 
x^2 + 2*x^3))*Log[5 + x]))/(5*x^3 + x^4 + (-10*x^2 - 2*x^3)*Log[5 + x] + ( 
5*x + x^2)*Log[5 + 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 = 0.32 (sec) , antiderivative size = 599, normalized size of antiderivative = 20.66

\[\frac {16+16 x -16 x \,{\mathrm e}^{x} \ln \left (x \right )+16 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (x \right )+32 \ln \left (x \right )-16 \,{\mathrm e}^{x}+4 \,{\mathrm e}^{2 x}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+4 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}-8 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-8 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (x \right )+16 \ln \left (x \right )^{2}-8 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} x +4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right )-6 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2}+4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4}+4 x \,{\mathrm e}^{2 x}-16 \,{\mathrm e}^{x} \ln \left (x \right )+16 x \ln \left (x \right )^{2}-16 \,{\mathrm e}^{x} x +32 x \ln \left (x \right )-8 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}-8 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )+4 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-8 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-8 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}+4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) x -6 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} x +4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} x -\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} x}{4 x -4 \ln \left (5+x \right )}\]

Input:

int((((-x^2-5*x)*ln(5+x)-4*x)*ln(x^2)^2+(((2*x^3+14*x^2+20*x)*exp(x)-8*x^2 
-44*x-20)*ln(5+x)+(-2*x^4-12*x^3-10*x^2+8*x)*exp(x)+4*x^3+24*x^2+4*x)*ln(x 
^2)+((-2*x^3-13*x^2-15*x)*exp(x)^2+(4*x^3+32*x^2+64*x+20)*exp(x)-12*x^2-68 
*x-40)*ln(5+x)+(2*x^4+12*x^3+10*x^2-4*x)*exp(x)^2+(-4*x^4-28*x^3-44*x^2-4* 
x)*exp(x)+8*x^3+48*x^2+24*x)/((x^2+5*x)*ln(5+x)^2+(-2*x^3-10*x^2)*ln(5+x)+ 
x^4+5*x^3),x)
 

Output:

1/4*(16+16*x-16*x*exp(x)*ln(x)-8*I*Pi*x*csgn(I*x^2)^3-8*I*Pi*csgn(I*x)^2*c 
sgn(I*x^2)+16*I*Pi*csgn(I*x)*csgn(I*x^2)^2-8*I*ln(x)*Pi*csgn(I*x^2)^3+4*I* 
Pi*csgn(I*x^2)^3*exp(x)+32*ln(x)-16*exp(x)+4*exp(x)^2+16*ln(x)^2-16*exp(x) 
*ln(x)+16*x*ln(x)^2+4*x*exp(x)^2-16*exp(x)*x+32*x*ln(x)-Pi^2*csgn(I*x^2)^6 
+4*Pi^2*csgn(I*x^2)^5*csgn(I*x)*x-6*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2*x+4*Pi^ 
2*csgn(I*x^2)^3*csgn(I*x)^3*x-Pi^2*csgn(I*x^2)^2*csgn(I*x)^4*x-8*I*Pi*x*cs 
gn(I*x)^2*csgn(I*x^2)+16*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2+4*I*Pi*csgn(I*x)^2 
*csgn(I*x^2)*exp(x)-8*I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(x)-8*I*ln(x)*Pi*csg 
n(I*x)^2*csgn(I*x^2)+16*I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)^2-8*I*Pi*x*csgn(I 
*x^2)^3*ln(x)+4*I*Pi*x*csgn(I*x^2)^3*exp(x)-8*I*Pi*csgn(I*x^2)^3-Pi^2*csgn 
(I*x^2)^6*x+4*Pi^2*csgn(I*x^2)^5*csgn(I*x)-6*Pi^2*csgn(I*x^2)^4*csgn(I*x)^ 
2+4*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3-Pi^2*csgn(I*x^2)^2*csgn(I*x)^4+4*I*Pi*x 
*csgn(I*x)^2*csgn(I*x^2)*exp(x)-8*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2*exp(x)-8* 
I*Pi*x*csgn(I*x)^2*csgn(I*x^2)*ln(x)+16*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2*ln( 
x))/(x-ln(5+x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=\frac {{\left (x + 1\right )} \log \left (x^{2}\right )^{2} + {\left (x + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x + 1\right )} e^{x} - 2 \, {\left ({\left (x + 1\right )} e^{x} - 2 \, x - 2\right )} \log \left (x^{2}\right ) + 4 \, x + 4}{x - \log \left (x + 5\right )} \] Input:

integrate((((-x^2-5*x)*log(5+x)-4*x)*log(x^2)^2+(((2*x^3+14*x^2+20*x)*exp( 
x)-8*x^2-44*x-20)*log(5+x)+(-2*x^4-12*x^3-10*x^2+8*x)*exp(x)+4*x^3+24*x^2+ 
4*x)*log(x^2)+((-2*x^3-13*x^2-15*x)*exp(x)^2+(4*x^3+32*x^2+64*x+20)*exp(x) 
-12*x^2-68*x-40)*log(5+x)+(2*x^4+12*x^3+10*x^2-4*x)*exp(x)^2+(-4*x^4-28*x^ 
3-44*x^2-4*x)*exp(x)+8*x^3+48*x^2+24*x)/((x^2+5*x)*log(5+x)^2+(-2*x^3-10*x 
^2)*log(5+x)+x^4+5*x^3),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.34 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=\frac {\left (x^{2} - x \log {\left (x + 5 \right )} + x - \log {\left (x + 5 \right )}\right ) e^{2 x} + \left (- 2 x^{2} \log {\left (x^{2} \right )} - 4 x^{2} + 2 x \log {\left (x^{2} \right )} \log {\left (x + 5 \right )} - 2 x \log {\left (x^{2} \right )} + 4 x \log {\left (x + 5 \right )} - 4 x + 2 \log {\left (x^{2} \right )} \log {\left (x + 5 \right )} + 4 \log {\left (x + 5 \right )}\right ) e^{x}}{x^{2} - 2 x \log {\left (x + 5 \right )} + \log {\left (x + 5 \right )}^{2}} + \frac {- x \log {\left (x^{2} \right )}^{2} - 4 x \log {\left (x^{2} \right )} - 4 x - \log {\left (x^{2} \right )}^{2} - 4 \log {\left (x^{2} \right )} - 4}{- x + \log {\left (x + 5 \right )}} \] Input:

integrate((((-x**2-5*x)*ln(5+x)-4*x)*ln(x**2)**2+(((2*x**3+14*x**2+20*x)*e 
xp(x)-8*x**2-44*x-20)*ln(5+x)+(-2*x**4-12*x**3-10*x**2+8*x)*exp(x)+4*x**3+ 
24*x**2+4*x)*ln(x**2)+((-2*x**3-13*x**2-15*x)*exp(x)**2+(4*x**3+32*x**2+64 
*x+20)*exp(x)-12*x**2-68*x-40)*ln(5+x)+(2*x**4+12*x**3+10*x**2-4*x)*exp(x) 
**2+(-4*x**4-28*x**3-44*x**2-4*x)*exp(x)+8*x**3+48*x**2+24*x)/((x**2+5*x)* 
ln(5+x)**2+(-2*x**3-10*x**2)*ln(5+x)+x**4+5*x**3),x)
 

Output:

((x**2 - x*log(x + 5) + x - log(x + 5))*exp(2*x) + (-2*x**2*log(x**2) - 4* 
x**2 + 2*x*log(x**2)*log(x + 5) - 2*x*log(x**2) + 4*x*log(x + 5) - 4*x + 2 
*log(x**2)*log(x + 5) + 4*log(x + 5))*exp(x))/(x**2 - 2*x*log(x + 5) + log 
(x + 5)**2) + (-x*log(x**2)**2 - 4*x*log(x**2) - 4*x - log(x**2)**2 - 4*lo 
g(x**2) - 4)/(-x + log(x + 5))
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=\frac {4 \, {\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left ({\left (x + 1\right )} \log \left (x\right ) + x + 1\right )} e^{x} + 8 \, {\left (x + 1\right )} \log \left (x\right ) + 4 \, x + 4}{x - \log \left (x + 5\right )} \] Input:

integrate((((-x^2-5*x)*log(5+x)-4*x)*log(x^2)^2+(((2*x^3+14*x^2+20*x)*exp( 
x)-8*x^2-44*x-20)*log(5+x)+(-2*x^4-12*x^3-10*x^2+8*x)*exp(x)+4*x^3+24*x^2+ 
4*x)*log(x^2)+((-2*x^3-13*x^2-15*x)*exp(x)^2+(4*x^3+32*x^2+64*x+20)*exp(x) 
-12*x^2-68*x-40)*log(5+x)+(2*x^4+12*x^3+10*x^2-4*x)*exp(x)^2+(-4*x^4-28*x^ 
3-44*x^2-4*x)*exp(x)+8*x^3+48*x^2+24*x)/((x^2+5*x)*log(5+x)^2+(-2*x^3-10*x 
^2)*log(5+x)+x^4+5*x^3),x, algorithm="maxima")
 

Output:

(4*(x + 1)*log(x)^2 + (x + 1)*e^(2*x) - 4*((x + 1)*log(x) + x + 1)*e^x + 8 
*(x + 1)*log(x) + 4*x + 4)/(x - log(x + 5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (28) = 56\).

Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=-\frac {2 \, x e^{x} \log \left (x^{2}\right ) - x \log \left (x^{2}\right )^{2} - x e^{\left (2 \, x\right )} + 4 \, x e^{x} - 4 \, x \log \left (x^{2}\right ) + 2 \, e^{x} \log \left (x^{2}\right ) - \log \left (x^{2}\right )^{2} - 4 \, x - e^{\left (2 \, x\right )} + 4 \, e^{x} - 4 \, \log \left (x^{2}\right ) - 4}{x - \log \left (x + 5\right )} \] Input:

integrate((((-x^2-5*x)*log(5+x)-4*x)*log(x^2)^2+(((2*x^3+14*x^2+20*x)*exp( 
x)-8*x^2-44*x-20)*log(5+x)+(-2*x^4-12*x^3-10*x^2+8*x)*exp(x)+4*x^3+24*x^2+ 
4*x)*log(x^2)+((-2*x^3-13*x^2-15*x)*exp(x)^2+(4*x^3+32*x^2+64*x+20)*exp(x) 
-12*x^2-68*x-40)*log(5+x)+(2*x^4+12*x^3+10*x^2-4*x)*exp(x)^2+(-4*x^4-28*x^ 
3-44*x^2-4*x)*exp(x)+8*x^3+48*x^2+24*x)/((x^2+5*x)*log(5+x)^2+(-2*x^3-10*x 
^2)*log(5+x)+x^4+5*x^3),x, algorithm="giac")
 

Output:

-(2*x*e^x*log(x^2) - x*log(x^2)^2 - x*e^(2*x) + 4*x*e^x - 4*x*log(x^2) + 2 
*e^x*log(x^2) - log(x^2)^2 - 4*x - e^(2*x) + 4*e^x - 4*log(x^2) - 4)/(x - 
log(x + 5))
 

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 743, normalized size of antiderivative = 25.62 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx =\text {Too large to display} \] Input:

int((24*x - exp(x)*(4*x + 44*x^2 + 28*x^3 + 4*x^4) - log(x + 5)*(68*x + ex 
p(2*x)*(15*x + 13*x^2 + 2*x^3) + 12*x^2 - exp(x)*(64*x + 32*x^2 + 4*x^3 + 
20) + 40) + log(x^2)*(4*x - exp(x)*(10*x^2 - 8*x + 12*x^3 + 2*x^4) + 24*x^ 
2 + 4*x^3 - log(x + 5)*(44*x + 8*x^2 - exp(x)*(20*x + 14*x^2 + 2*x^3) + 20 
)) + exp(2*x)*(10*x^2 - 4*x + 12*x^3 + 2*x^4) + 48*x^2 + 8*x^3 - log(x^2)^ 
2*(4*x + log(x + 5)*(5*x + x^2)))/(log(x + 5)^2*(5*x + x^2) - log(x + 5)*( 
10*x^2 + 2*x^3) + 5*x^3 + x^4),x)
 

Output:

16*log(x) + 4/(x + 4) + ((2*(2*exp(2*x) - 8*exp(x) - 5*x*exp(2*x) + 12*x^2 
*exp(x) + 2*x^3*exp(x) - 6*x^2*exp(2*x) - x^3*exp(2*x) + 10*x*exp(x) + 8)) 
/(x + 4) + (log(x + 5)*(x + 5)*(3*exp(2*x) - 8*exp(x) + 2*x*exp(2*x) - 4*x 
*exp(x) + 4))/(x + 4))/(x - log(x + 5)) + log(x^2)^2*((4/(x + 4) + log(x + 
 5)*((x + 5)/(x + 4) - (20*x + 9*x^2 + x^3)/(x*(x + 4)^2*(x + 5))) + (20*x 
 + 9*x^2 + x^3)/((x + 4)^2*(x + 5)))/(x - log(x + 5)) + 1) + (exp(2*x)*(13 
*x + 2*x^2 + 15))/(x + 4) + (log(x^2)*((2*(6*x^2*exp(x) - 4*exp(x) - 10*x 
+ x^3*exp(x) + 5*x*exp(x) - 2*x^2 + 8))/(x + 4) - x*((80*x + 36*x^2 + 4*x^ 
3)/(x*(x + 4)^2*(x + 5)) - (2*(320*x + 184*x^2 + 34*x^3 + 2*x^4))/(x*(x + 
4)^2*(x + 5)) + (400*x + 260*x^2 + 56*x^3 + 4*x^4)/(x*(x + 4)^2*(x + 5)) - 
 (exp(x)*(80*x + 36*x^2 + 4*x^3))/(x*(x + 4)^2*(x + 5)) + (exp(x)*(480*x + 
 496*x^2 + 190*x^3 + 32*x^4 + 2*x^5))/(x*(x + 4)^2*(x + 5))) + log(x + 5)^ 
2*((80*x + 36*x^2 + 4*x^3)/(x^2*(x + 4)^2) - (4*(x + 5))/(x*(x + 4))) - lo 
g(x + 5)*((80*x + 36*x^2 + 4*x^3)/(x*(x + 4)^2) + (2*(x + 5)*(2*exp(x) + x 
*exp(x) - 6))/(x + 4) - (80*x + 36*x^2 + 4*x^3)/(x*(x + 4)^2*(x + 5)) + (2 
*(320*x + 184*x^2 + 34*x^3 + 2*x^4))/(x*(x + 4)^2*(x + 5)) - (400*x + 260* 
x^2 + 56*x^3 + 4*x^4)/(x*(x + 4)^2*(x + 5)) + (exp(x)*(80*x + 36*x^2 + 4*x 
^3))/(x*(x + 4)^2*(x + 5)) - (exp(x)*(480*x + 496*x^2 + 190*x^3 + 32*x^4 + 
 2*x^5))/(x*(x + 4)^2*(x + 5)))))/(x - log(x + 5)) - (exp(x)*(28*x + 4*x^2 
 + 40))/(x + 4)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.79 \[ \int \frac {24 x+48 x^2+8 x^3+e^x \left (-4 x-44 x^2-28 x^3-4 x^4\right )+e^{2 x} \left (-4 x+10 x^2+12 x^3+2 x^4\right )+\left (-40-68 x-12 x^2+e^{2 x} \left (-15 x-13 x^2-2 x^3\right )+e^x \left (20+64 x+32 x^2+4 x^3\right )\right ) \log (5+x)+\log ^2\left (x^2\right ) \left (-4 x+\left (-5 x-x^2\right ) \log (5+x)\right )+\log \left (x^2\right ) \left (4 x+24 x^2+4 x^3+e^x \left (8 x-10 x^2-12 x^3-2 x^4\right )+\left (-20-44 x-8 x^2+e^x \left (20 x+14 x^2+2 x^3\right )\right ) \log (5+x)\right )}{5 x^3+x^4+\left (-10 x^2-2 x^3\right ) \log (5+x)+\left (5 x+x^2\right ) \log ^2(5+x)} \, dx=\frac {-e^{2 x} x -e^{2 x}+2 e^{x} \mathrm {log}\left (x^{2}\right ) x +2 e^{x} \mathrm {log}\left (x^{2}\right )+4 e^{x} x +4 e^{x}-\mathrm {log}\left (x^{2}\right )^{2} x -\mathrm {log}\left (x^{2}\right )^{2}-4 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x +5\right )-4 \,\mathrm {log}\left (x^{2}\right )+8 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right )-4 \,\mathrm {log}\left (x +5\right )-8 \,\mathrm {log}\left (x \right ) x -4}{\mathrm {log}\left (x +5\right )-x} \] Input:

int((((-x^2-5*x)*log(5+x)-4*x)*log(x^2)^2+(((2*x^3+14*x^2+20*x)*exp(x)-8*x 
^2-44*x-20)*log(5+x)+(-2*x^4-12*x^3-10*x^2+8*x)*exp(x)+4*x^3+24*x^2+4*x)*l 
og(x^2)+((-2*x^3-13*x^2-15*x)*exp(x)^2+(4*x^3+32*x^2+64*x+20)*exp(x)-12*x^ 
2-68*x-40)*log(5+x)+(2*x^4+12*x^3+10*x^2-4*x)*exp(x)^2+(-4*x^4-28*x^3-44*x 
^2-4*x)*exp(x)+8*x^3+48*x^2+24*x)/((x^2+5*x)*log(5+x)^2+(-2*x^3-10*x^2)*lo 
g(5+x)+x^4+5*x^3),x)
 

Output:

( - e**(2*x)*x - e**(2*x) + 2*e**x*log(x**2)*x + 2*e**x*log(x**2) + 4*e**x 
*x + 4*e**x - log(x**2)**2*x - log(x**2)**2 - 4*log(x**2)*log(x + 5) - 4*l 
og(x**2) + 8*log(x + 5)*log(x) - 4*log(x + 5) - 8*log(x)*x - 4)/(log(x + 5 
) - x)