\(\int \frac {e^{\frac {(1152+144 e^x) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} ((-4608 x-576 e^x x) \log ^3(x)+(4608 x+e^x (576 x-144 x^2)) \log ^4(x)+(-2304+e^x (-288+144 x)) \log ^6(x))}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx\) [1325]

Optimal result

Integrand size = 130, antiderivative size = 25 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\frac {144 \left (8+e^x\right )}{5 \left (-x+\frac {x^2}{\log ^2(x)}\right )^2}} \] Output:

exp(144/5/(x^2/ln(x)^2-x)^2*(exp(x)+8))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}} \] Input:

Integrate[(E^(((1152 + 144*E^x)*Log[x]^4)/(5*x^4 - 10*x^3*Log[x]^2 + 5*x^2 
*Log[x]^4))*((-4608*x - 576*E^x*x)*Log[x]^3 + (4608*x + E^x*(576*x - 144*x 
^2))*Log[x]^4 + (-2304 + E^x*(-288 + 144*x))*Log[x]^6))/(-5*x^6 + 15*x^5*L 
og[x]^2 - 15*x^4*Log[x]^4 + 5*x^3*Log[x]^6),x]
 

Output:

E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^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.

\(\displaystyle \int \frac {\left (\left (e^x \left (576 x-144 x^2\right )+4608 x\right ) \log ^4(x)+\left (e^x (144 x-288)-2304\right ) \log ^6(x)+\left (-576 e^x x-4608 x\right ) \log ^3(x)\right ) \exp \left (\frac {\left (144 e^x+1152\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-\left (e^x \left (576 x-144 x^2\right )+4608 x\right ) \log ^4(x)-\left (\left (e^x (144 x-288)-2304\right ) \log ^6(x)\right )-\left (-576 e^x x-4608 x\right ) \log ^3(x)\right ) \exp \left (\frac {144 \left (e^x+8\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{5 x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {144 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \left (\left (e^x (2-x)+16\right ) \log ^6(x)-\left (32 x+e^x \left (4 x-x^2\right )\right ) \log ^4(x)+4 \left (e^x x+8 x\right ) \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \left (\left (e^x (2-x)+16\right ) \log ^6(x)-\left (32 x+e^x \left (4 x-x^2\right )\right ) \log ^4(x)+4 \left (e^x x+8 x\right ) \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (\left (16-e^x (x-2)\right ) \log ^3(x)+\left (e^x (x-4)-32\right ) x \log (x)+4 \left (8+e^x\right ) x\right )}{x^3 \left (x-\log ^2(x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {144}{5} \int \left (\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}+x\right ) \left (-x \log ^3(x)+2 \log ^3(x)+x^2 \log (x)-4 x \log (x)+4 x\right ) \log ^3(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}\right )dx\)

Input:

Int[(E^(((1152 + 144*E^x)*Log[x]^4)/(5*x^4 - 10*x^3*Log[x]^2 + 5*x^2*Log[x 
]^4))*((-4608*x - 576*E^x*x)*Log[x]^3 + (4608*x + E^x*(576*x - 144*x^2))*L 
og[x]^4 + (-2304 + E^x*(-288 + 144*x))*Log[x]^6))/(-5*x^6 + 15*x^5*Log[x]^ 
2 - 15*x^4*Log[x]^4 + 5*x^3*Log[x]^6),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[{\mathrm e}^{\frac {144 \left ({\mathrm e}^{x}+8\right ) \ln \left (x \right )^{4}}{5 x^{2} \left (\ln \left (x \right )^{2}-x \right )^{2}}}\]

Input:

int((((144*x-288)*exp(x)-2304)*ln(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*ln 
(x)^4+(-576*exp(x)*x-4608*x)*ln(x)^3)*exp((144*exp(x)+1152)*ln(x)^4/(5*x^2 
*ln(x)^4-10*x^3*ln(x)^2+5*x^4))/(5*x^3*ln(x)^6-15*x^4*ln(x)^4+15*x^5*ln(x) 
^2-5*x^6),x)
 

Output:

exp(144/5*(exp(x)+8)*ln(x)^4/x^2/(ln(x)^2-x)^2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\left (\frac {144 \, {\left (e^{x} + 8\right )} \log \left (x\right )^{4}}{5 \, {\left (x^{2} \log \left (x\right )^{4} - 2 \, x^{3} \log \left (x\right )^{2} + x^{4}\right )}}\right )} \] Input:

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+460 
8*x)*log(x)^4+(-576*exp(x)*x-4608*x)*log(x)^3)*exp((144*exp(x)+1152)*log(x 
)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x)^ 
4+15*x^5*log(x)^2-5*x^6),x, algorithm="fricas")
 

Output:

e^(144/5*(e^x + 8)*log(x)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 + x^4))
 

Sympy [A] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\frac {\left (144 e^{x} + 1152\right ) \log {\left (x \right )}^{4}}{5 x^{4} - 10 x^{3} \log {\left (x \right )}^{2} + 5 x^{2} \log {\left (x \right )}^{4}}} \] Input:

integrate((((144*x-288)*exp(x)-2304)*ln(x)**6+((-144*x**2+576*x)*exp(x)+46 
08*x)*ln(x)**4+(-576*exp(x)*x-4608*x)*ln(x)**3)*exp((144*exp(x)+1152)*ln(x 
)**4/(5*x**2*ln(x)**4-10*x**3*ln(x)**2+5*x**4))/(5*x**3*ln(x)**6-15*x**4*l 
n(x)**4+15*x**5*ln(x)**2-5*x**6),x)
 

Output:

exp((144*exp(x) + 1152)*log(x)**4/(5*x**4 - 10*x**3*log(x)**2 + 5*x**2*log 
(x)**4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (20) = 40\).

Time = 3.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\left (\frac {288 \, e^{x}}{5 \, {\left (\log \left (x\right )^{4} - x \log \left (x\right )^{2}\right )}} + \frac {144 \, e^{x}}{5 \, {\left (\log \left (x\right )^{4} - 2 \, x \log \left (x\right )^{2} + x^{2}\right )}} + \frac {2304}{5 \, {\left (\log \left (x\right )^{4} - x \log \left (x\right )^{2}\right )}} + \frac {1152}{5 \, {\left (\log \left (x\right )^{4} - 2 \, x \log \left (x\right )^{2} + x^{2}\right )}} + \frac {144 \, e^{x}}{5 \, x^{2}} + \frac {1152}{5 \, x^{2}} + \frac {288 \, e^{x}}{5 \, x \log \left (x\right )^{2}} + \frac {2304}{5 \, x \log \left (x\right )^{2}}\right )} \] Input:

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+460 
8*x)*log(x)^4+(-576*exp(x)*x-4608*x)*log(x)^3)*exp((144*exp(x)+1152)*log(x 
)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x)^ 
4+15*x^5*log(x)^2-5*x^6),x, algorithm="maxima")
 

Output:

e^(288/5*e^x/(log(x)^4 - x*log(x)^2) + 144/5*e^x/(log(x)^4 - 2*x*log(x)^2 
+ x^2) + 2304/5/(log(x)^4 - x*log(x)^2) + 1152/5/(log(x)^4 - 2*x*log(x)^2 
+ x^2) + 144/5*e^x/x^2 + 1152/5/x^2 + 288/5*e^x/(x*log(x)^2) + 2304/5/(x*l 
og(x)^2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (20) = 40\).

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\left (\frac {144 \, e^{x} \log \left (x\right )^{4}}{5 \, {\left (x^{2} \log \left (x\right )^{4} - 2 \, x^{3} \log \left (x\right )^{2} + x^{4}\right )}} + \frac {1152 \, \log \left (x\right )^{4}}{5 \, {\left (x^{2} \log \left (x\right )^{4} - 2 \, x^{3} \log \left (x\right )^{2} + x^{4}\right )}}\right )} \] Input:

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+460 
8*x)*log(x)^4+(-576*exp(x)*x-4608*x)*log(x)^3)*exp((144*exp(x)+1152)*log(x 
)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x)^ 
4+15*x^5*log(x)^2-5*x^6),x, algorithm="giac")
 

Output:

e^(144/5*e^x*log(x)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 + x^4) + 1152/5*log(x 
)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 + x^4))
 

Mupad [B] (verification not implemented)

Time = 7.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx={\mathrm {e}}^{\frac {1152\,{\ln \left (x\right )}^4+144\,{\mathrm {e}}^x\,{\ln \left (x\right )}^4}{5\,x^4-10\,x^3\,{\ln \left (x\right )}^2+5\,x^2\,{\ln \left (x\right )}^4}} \] Input:

int((exp((log(x)^4*(144*exp(x) + 1152))/(5*x^2*log(x)^4 - 10*x^3*log(x)^2 
+ 5*x^4))*(log(x)^6*(exp(x)*(144*x - 288) - 2304) + log(x)^4*(4608*x + exp 
(x)*(576*x - 144*x^2)) - log(x)^3*(4608*x + 576*x*exp(x))))/(15*x^5*log(x) 
^2 - 15*x^4*log(x)^4 + 5*x^3*log(x)^6 - 5*x^6),x)
 

Output:

exp((1152*log(x)^4 + 144*exp(x)*log(x)^4)/(5*x^2*log(x)^4 - 10*x^3*log(x)^ 
2 + 5*x^4))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {e^{\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx=e^{\frac {144 e^{x} \mathrm {log}\left (x \right )^{4}+1152 \mathrm {log}\left (x \right )^{4}}{5 \mathrm {log}\left (x \right )^{4} x^{2}-10 \mathrm {log}\left (x \right )^{2} x^{3}+5 x^{4}}} \] Input:

int((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*l 
og(x)^4+(-576*exp(x)*x-4608*x)*log(x)^3)*exp((144*exp(x)+1152)*log(x)^4/(5 
*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x)^4+15*x 
^5*log(x)^2-5*x^6),x)
 

Output:

e**((144*e**x*log(x)**4 + 1152*log(x)**4)/(5*log(x)**4*x**2 - 10*log(x)**2 
*x**3 + 5*x**4))