3.11.35 \(\int e^{-2 x+e^{-2 x} (x^2+2 e^x x^3+e^{2 x} x^4+(-10 x^2-2 x^3+e^x (-10 x^3-2 x^4)) \log ^2(x)+(25 x^2+10 x^3+x^4) \log ^4(x))} (2 x-2 x^2+4 e^{2 x} x^3+e^x (6 x^2-2 x^3)+(-20 x-4 x^2+e^x (-20 x^2-4 x^3)) \log (x)+(-20 x+14 x^2+4 x^3+e^x (-30 x^2+2 x^3+2 x^4)) \log ^2(x)+(100 x+40 x^2+4 x^3) \log ^3(x)+(50 x-20 x^2-16 x^3-2 x^4) \log ^4(x)) \, dx\) [1035]

3.11.35.1 Optimal result

Integrand size = 227, antiderivative size = 27 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=e^{x^2 \left (x-e^{-x} \left (-1+(5+x) \log ^2(x)\right )\right )^2} \end {dmath*}

output

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

3.11.35.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=e^{e^{-2 x} x^2 \left (1+e^x x-(5+x) \log ^2(x)\right )^2} \end {dmath*}

input

Integrate[E^(-2*x + (x^2 + 2*E^x*x^3 + E^(2*x)*x^4 + (-10*x^2 - 2*x^3 + E^ 
x*(-10*x^3 - 2*x^4))*Log[x]^2 + (25*x^2 + 10*x^3 + x^4)*Log[x]^4)/E^(2*x)) 
*(2*x - 2*x^2 + 4*E^(2*x)*x^3 + E^x*(6*x^2 - 2*x^3) + (-20*x - 4*x^2 + E^x 
*(-20*x^2 - 4*x^3))*Log[x] + (-20*x + 14*x^2 + 4*x^3 + E^x*(-30*x^2 + 2*x^ 
3 + 2*x^4))*Log[x]^2 + (100*x + 40*x^2 + 4*x^3)*Log[x]^3 + (50*x - 20*x^2 
- 16*x^3 - 2*x^4)*Log[x]^4),x]
 

output

E^((x^2*(1 + E^x*x - (5 + x)*Log[x]^2)^2)/E^(2*x))
 

3.11.35.3 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.

\(\displaystyle \int \left (4 e^{2 x} x^3-2 x^2+e^x \left (6 x^2-2 x^3\right )+\left (4 x^3+40 x^2+100 x\right ) \log ^3(x)+\left (-4 x^2+e^x \left (-4 x^3-20 x^2\right )-20 x\right ) \log (x)+\left (-2 x^4-16 x^3-20 x^2+50 x\right ) \log ^4(x)+\left (4 x^3+14 x^2+e^x \left (2 x^4+2 x^3-30 x^2\right )-20 x\right ) \log ^2(x)+2 x\right ) \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right ) \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-2 x \left (x^3+8 x^2+10 x-25\right ) \log ^4(x) \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )+2 x \left (e^x x^3+e^x x^2+2 x^2-15 e^x x+7 x-10\right ) \log ^2(x) \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )-4 x (x+5) \left (e^x x+1\right ) \log (x) \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )+4 x^3 \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )\right )-2 x^2 \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )-2 (x-3) x^2 \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-x\right )+2 x \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )+4 x (x+5)^2 \log ^3(x) \exp \left (e^{-2 x} \left (e^{2 x} x^4+2 e^x x^3+x^2+\left (x^4+10 x^3+25 x^2\right ) \log ^4(x)+\left (-2 x^3-10 x^2+e^x \left (-2 x^4-10 x^3\right )\right ) \log ^2(x)\right )-2 x\right )\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int 2 x \left (e^x \left (2 e^x-1\right ) x^2-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+\left (e^x x^3+\left (e^x+2\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)+\left (3 e^x-1\right ) x+2 (x+5)^2 \log ^3(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+1\right ) \exp \left (e^{-2 x} x \left (e^{2 x} \left (x^3-2\right )+2 e^x x^2+x+x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)\right )\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x-e^{2 x} \left (2-x^3\right )\right )\right ) x \left (\left (-x^3-8 x^2-10 x+25\right ) \log ^4(x)+2 (x+5)^2 \log ^3(x)-\left (-e^x x^3-\left (2+e^x\right ) x^2-\left (7-15 e^x\right ) x+10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)-e^x \left (1-2 e^x\right ) x^2-\left (1-3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x-e^{2 x} \left (2-x^3\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x-e^{2 x} \left (2-x^3\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x-e^{2 x} \left (2-x^3\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) x \left (-\left (\left (x^3+8 x^2+10 x-25\right ) \log ^4(x)\right )+2 (x+5)^2 \log ^3(x)+\left (e^x x^3+\left (2+e^x\right ) x^2+\left (7-15 e^x\right ) x-10\right ) \log ^2(x)-2 (x+5) \left (e^x x+1\right ) \log (x)+e^x \left (-1+2 e^x\right ) x^2+\left (-1+3 e^x\right ) x+1\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+2 x\right ) x^3+\exp \left (e^{-2 x} \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right ) x+x\right ) \left (x^2 \log ^2(x)+x \log ^2(x)-15 \log ^2(x)-2 x \log (x)-10 \log (x)-x+3\right ) x^2-\exp \left (e^{-2 x} x \left (x (x+5)^2 \log ^4(x)-2 x (x+5) \left (e^x x+1\right ) \log ^2(x)+2 e^x x^2+x+e^{2 x} \left (x^3-2\right )\right )\right ) \left (x^3 \log ^4(x)+8 x^2 \log ^4(x)+10 x \log ^4(x)-25 \log ^4(x)-2 x^2 \log ^3(x)-20 x \log ^3(x)-50 \log ^3(x)-2 x^2 \log ^2(x)-7 x \log ^2(x)+10 \log ^2(x)+2 x \log (x)+10 \log (x)+x-1\right ) x\right )dx\)

input

Int[E^(-2*x + (x^2 + 2*E^x*x^3 + E^(2*x)*x^4 + (-10*x^2 - 2*x^3 + E^x*(-10 
*x^3 - 2*x^4))*Log[x]^2 + (25*x^2 + 10*x^3 + x^4)*Log[x]^4)/E^(2*x))*(2*x 
- 2*x^2 + 4*E^(2*x)*x^3 + E^x*(6*x^2 - 2*x^3) + (-20*x - 4*x^2 + E^x*(-20* 
x^2 - 4*x^3))*Log[x] + (-20*x + 14*x^2 + 4*x^3 + E^x*(-30*x^2 + 2*x^3 + 2* 
x^4))*Log[x]^2 + (100*x + 40*x^2 + 4*x^3)*Log[x]^3 + (50*x - 20*x^2 - 16*x 
^3 - 2*x^4)*Log[x]^4),x]
 

output

$Aborted
 

3.11.35.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.11.35.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(25)=50\).

Time = 5.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78

method	result	size
parallelrisch	\({\mathrm e}^{\left (\left (x^{4}+10 x^{3}+25 x^{2}\right ) \ln \left (x \right )^{4}+\left (\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}^{x}-2 x^{3}-10 x^{2}\right ) \ln \left (x \right )^{2}+{\mathrm e}^{2 x} x^{4}+2 \,{\mathrm e}^{x} x^{3}+x^{2}\right ) {\mathrm e}^{-2 x}}\)	\(75\)
risch	\({\mathrm e}^{-x^{2} \left (-x^{2} \ln \left (x \right )^{4}-10 x \ln \left (x \right )^{4}+2 x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}-25 \ln \left (x \right )^{4}+10 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+2 x \ln \left (x \right )^{2}-{\mathrm e}^{2 x} x^{2}+10 \ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} x -1\right ) {\mathrm e}^{-2 x}}\)	\(82\)

input

int(((-2*x^4-16*x^3-20*x^2+50*x)*ln(x)^4+(4*x^3+40*x^2+100*x)*ln(x)^3+((2* 
x^4+2*x^3-30*x^2)*exp(x)+4*x^3+14*x^2-20*x)*ln(x)^2+((-4*x^3-20*x^2)*exp(x 
)-4*x^2-20*x)*ln(x)+4*exp(x)^2*x^3+(-2*x^3+6*x^2)*exp(x)-2*x^2+2*x)*exp((( 
x^4+10*x^3+25*x^2)*ln(x)^4+((-2*x^4-10*x^3)*exp(x)-2*x^3-10*x^2)*ln(x)^2+e 
xp(x)^2*x^4+2*exp(x)*x^3+x^2)/exp(x)^2)/exp(x)^2,x,method=_RETURNVERBOSE)
 

output

exp(((x^4+10*x^3+25*x^2)*ln(x)^4+((-2*x^4-10*x^3)*exp(x)-2*x^3-10*x^2)*ln( 
x)^2+exp(x)^2*x^4+2*exp(x)*x^3+x^2)/exp(x)^2)
 

3.11.35.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=e^{\left ({\left ({\left (x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )} \log \left (x\right )^{4} + 2 \, x^{3} e^{x} - 2 \, {\left (x^{3} + 5 \, x^{2} + {\left (x^{4} + 5 \, x^{3}\right )} e^{x}\right )} \log \left (x\right )^{2} + x^{2} + {\left (x^{4} - 2 \, x\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \end {dmath*}

input

integrate(((-2*x^4-16*x^3-20*x^2+50*x)*log(x)^4+(4*x^3+40*x^2+100*x)*log(x 
)^3+((2*x^4+2*x^3-30*x^2)*exp(x)+4*x^3+14*x^2-20*x)*log(x)^2+((-4*x^3-20*x 
^2)*exp(x)-4*x^2-20*x)*log(x)+4*exp(x)^2*x^3+(-2*x^3+6*x^2)*exp(x)-2*x^2+2 
*x)*exp(((x^4+10*x^3+25*x^2)*log(x)^4+((-2*x^4-10*x^3)*exp(x)-2*x^3-10*x^2 
)*log(x)^2+exp(x)^2*x^4+2*exp(x)*x^3+x^2)/exp(x)^2)/exp(x)^2,x, algorithm= 
\
 

output

e^(((x^4 + 10*x^3 + 25*x^2)*log(x)^4 + 2*x^3*e^x - 2*(x^3 + 5*x^2 + (x^4 + 
 5*x^3)*e^x)*log(x)^2 + x^2 + (x^4 - 2*x)*e^(2*x))*e^(-2*x) + 2*x)
 

3.11.35.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).

Time = 1.00 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=e^{\left (x^{4} e^{2 x} + 2 x^{3} e^{x} + x^{2} + \left (- 2 x^{3} - 10 x^{2} + \left (- 2 x^{4} - 10 x^{3}\right ) e^{x}\right ) \log {\left (x \right )}^{2} + \left (x^{4} + 10 x^{3} + 25 x^{2}\right ) \log {\left (x \right )}^{4}\right ) e^{- 2 x}} \end {dmath*}

input

integrate(((-2*x**4-16*x**3-20*x**2+50*x)*ln(x)**4+(4*x**3+40*x**2+100*x)* 
ln(x)**3+((2*x**4+2*x**3-30*x**2)*exp(x)+4*x**3+14*x**2-20*x)*ln(x)**2+((- 
4*x**3-20*x**2)*exp(x)-4*x**2-20*x)*ln(x)+4*exp(x)**2*x**3+(-2*x**3+6*x**2 
)*exp(x)-2*x**2+2*x)*exp(((x**4+10*x**3+25*x**2)*ln(x)**4+((-2*x**4-10*x** 
3)*exp(x)-2*x**3-10*x**2)*ln(x)**2+exp(x)**2*x**4+2*exp(x)*x**3+x**2)/exp( 
x)**2)/exp(x)**2,x)
 

output

exp((x**4*exp(2*x) + 2*x**3*exp(x) + x**2 + (-2*x**3 - 10*x**2 + (-2*x**4 
- 10*x**3)*exp(x))*log(x)**2 + (x**4 + 10*x**3 + 25*x**2)*log(x)**4)*exp(- 
2*x))
 

3.11.35.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).

Time = 0.67 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=e^{\left (x^{4} e^{\left (-2 \, x\right )} \log \left (x\right )^{4} + 10 \, x^{3} e^{\left (-2 \, x\right )} \log \left (x\right )^{4} - 2 \, x^{4} e^{\left (-x\right )} \log \left (x\right )^{2} + 25 \, x^{2} e^{\left (-2 \, x\right )} \log \left (x\right )^{4} - 10 \, x^{3} e^{\left (-x\right )} \log \left (x\right )^{2} - 2 \, x^{3} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} - 10 \, x^{2} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} + x^{4} + 2 \, x^{3} e^{\left (-x\right )} + x^{2} e^{\left (-2 \, x\right )}\right )} \end {dmath*}

input

integrate(((-2*x^4-16*x^3-20*x^2+50*x)*log(x)^4+(4*x^3+40*x^2+100*x)*log(x 
)^3+((2*x^4+2*x^3-30*x^2)*exp(x)+4*x^3+14*x^2-20*x)*log(x)^2+((-4*x^3-20*x 
^2)*exp(x)-4*x^2-20*x)*log(x)+4*exp(x)^2*x^3+(-2*x^3+6*x^2)*exp(x)-2*x^2+2 
*x)*exp(((x^4+10*x^3+25*x^2)*log(x)^4+((-2*x^4-10*x^3)*exp(x)-2*x^3-10*x^2 
)*log(x)^2+exp(x)^2*x^4+2*exp(x)*x^3+x^2)/exp(x)^2)/exp(x)^2,x, algorithm= 
\
 

output

e^(x^4*e^(-2*x)*log(x)^4 + 10*x^3*e^(-2*x)*log(x)^4 - 2*x^4*e^(-x)*log(x)^ 
2 + 25*x^2*e^(-2*x)*log(x)^4 - 10*x^3*e^(-x)*log(x)^2 - 2*x^3*e^(-2*x)*log 
(x)^2 - 10*x^2*e^(-2*x)*log(x)^2 + x^4 + 2*x^3*e^(-x) + x^2*e^(-2*x))
 

3.11.35.8 Giac [F]

\begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx=\int { -2 \, {\left ({\left (x^{4} + 8 \, x^{3} + 10 \, x^{2} - 25 \, x\right )} \log \left (x\right )^{4} - 2 \, x^{3} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + 10 \, x^{2} + 25 \, x\right )} \log \left (x\right )^{3} - {\left (2 \, x^{3} + 7 \, x^{2} + {\left (x^{4} + x^{3} - 15 \, x^{2}\right )} e^{x} - 10 \, x\right )} \log \left (x\right )^{2} + x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} + 2 \, {\left (x^{2} + {\left (x^{3} + 5 \, x^{2}\right )} e^{x} + 5 \, x\right )} \log \left (x\right ) - x\right )} e^{\left ({\left (x^{4} e^{\left (2 \, x\right )} + {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )} \log \left (x\right )^{4} + 2 \, x^{3} e^{x} - 2 \, {\left (x^{3} + 5 \, x^{2} + {\left (x^{4} + 5 \, x^{3}\right )} e^{x}\right )} \log \left (x\right )^{2} + x^{2}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )} \,d x } \end {dmath*}

input

integrate(((-2*x^4-16*x^3-20*x^2+50*x)*log(x)^4+(4*x^3+40*x^2+100*x)*log(x 
)^3+((2*x^4+2*x^3-30*x^2)*exp(x)+4*x^3+14*x^2-20*x)*log(x)^2+((-4*x^3-20*x 
^2)*exp(x)-4*x^2-20*x)*log(x)+4*exp(x)^2*x^3+(-2*x^3+6*x^2)*exp(x)-2*x^2+2 
*x)*exp(((x^4+10*x^3+25*x^2)*log(x)^4+((-2*x^4-10*x^3)*exp(x)-2*x^3-10*x^2 
)*log(x)^2+exp(x)^2*x^4+2*exp(x)*x^3+x^2)/exp(x)^2)/exp(x)^2,x, algorithm= 
\
 

output

integrate(-2*((x^4 + 8*x^3 + 10*x^2 - 25*x)*log(x)^4 - 2*x^3*e^(2*x) - 2*( 
x^3 + 10*x^2 + 25*x)*log(x)^3 - (2*x^3 + 7*x^2 + (x^4 + x^3 - 15*x^2)*e^x 
- 10*x)*log(x)^2 + x^2 + (x^3 - 3*x^2)*e^x + 2*(x^2 + (x^3 + 5*x^2)*e^x + 
5*x)*log(x) - x)*e^((x^4*e^(2*x) + (x^4 + 10*x^3 + 25*x^2)*log(x)^4 + 2*x^ 
3*e^x - 2*(x^3 + 5*x^2 + (x^4 + 5*x^3)*e^x)*log(x)^2 + x^2)*e^(-2*x) - 2*x 
), x)
 

3.11.35.9 Mupad [B] (verification not implemented)

Time = 17.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.48 \begin {dmath*} \int e^{-2 x+e^{-2 x} \left (x^2+2 e^x x^3+e^{2 x} x^4+\left (-10 x^2-2 x^3+e^x \left (-10 x^3-2 x^4\right )\right ) \log ^2(x)+\left (25 x^2+10 x^3+x^4\right ) \log ^4(x)\right )} \left (2 x-2 x^2+4 e^{2 x} x^3+e^x \left (6 x^2-2 x^3\right )+\left (-20 x-4 x^2+e^x \left (-20 x^2-4 x^3\right )\right ) \log (x)+\left (-20 x+14 x^2+4 x^3+e^x \left (-30 x^2+2 x^3+2 x^4\right )\right ) \log ^2(x)+\left (100 x+40 x^2+4 x^3\right ) \log ^3(x)+\left (50 x-20 x^2-16 x^3-2 x^4\right ) \log ^4(x)\right ) \, dx={\mathrm {e}}^{-2\,x^3\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-2\,x^4\,{\mathrm {e}}^{-x}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{-10\,x^2\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{10\,x^3\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{25\,x^2\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2\,x}}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{-x}} \end {dmath*}

input

int(exp(-2*x)*exp(exp(-2*x)*(2*x^3*exp(x) + log(x)^4*(25*x^2 + 10*x^3 + x^ 
4) + x^4*exp(2*x) - log(x)^2*(exp(x)*(10*x^3 + 2*x^4) + 10*x^2 + 2*x^3) + 
x^2))*(2*x + exp(x)*(6*x^2 - 2*x^3) + log(x)^3*(100*x + 40*x^2 + 4*x^3) + 
4*x^3*exp(2*x) - log(x)^4*(20*x^2 - 50*x + 16*x^3 + 2*x^4) + log(x)^2*(exp 
(x)*(2*x^3 - 30*x^2 + 2*x^4) - 20*x + 14*x^2 + 4*x^3) - log(x)*(20*x + exp 
(x)*(20*x^2 + 4*x^3) + 4*x^2) - 2*x^2),x)
 

output

exp(-2*x^3*exp(-2*x)*log(x)^2)*exp(-2*x^4*exp(-x)*log(x)^2)*exp(x^4*exp(-2 
*x)*log(x)^4)*exp(-10*x^2*exp(-2*x)*log(x)^2)*exp(-10*x^3*exp(-x)*log(x)^2 
)*exp(10*x^3*exp(-2*x)*log(x)^4)*exp(25*x^2*exp(-2*x)*log(x)^4)*exp(x^4)*e 
xp(x^2*exp(-2*x))*exp(2*x^3*exp(-x))