\(\int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} (-4 x-48 x^2-32 x^3+34 x^4-6 x^5)+(-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)) \log (3-x)}{-3 x^2+x^3+(12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} (6 x-2 x^2)) \log (3-x)+(-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)) \log ^2(3-x)} \, dx\) [768]

Optimal result

Integrand size = 228, antiderivative size = 35 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {x}{-\frac {x}{2+e^{8 x \left (-2-x+\frac {x^2}{4}\right )}}+\log (3-x)} \] Output:

x/(ln(3-x)-x/(2+exp(x*(x^2-4*x-8))^2))
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {\left (e^{2 x^3}+2 e^{8 x (2+x)}\right ) x}{-e^{8 x (2+x)} x+\left (e^{2 x^3}+2 e^{8 x (2+x)}\right ) \log (3-x)} \] Input:

Integrate[(-4*x - E^(-32*x - 16*x^2 + 4*x^3)*x + E^(-16*x - 8*x^2 + 2*x^3) 
*(-4*x - 48*x^2 - 32*x^3 + 34*x^4 - 6*x^5) + (-12 + E^(-32*x - 16*x^2 + 4* 
x^3)*(-3 + x) + 4*x + E^(-16*x - 8*x^2 + 2*x^3)*(-12 + 4*x))*Log[3 - x])/( 
-3*x^2 + x^3 + (12*x - 4*x^2 + E^(-16*x - 8*x^2 + 2*x^3)*(6*x - 2*x^2))*Lo 
g[3 - x] + (-12 + E^(-32*x - 16*x^2 + 4*x^3)*(-3 + x) + 4*x + E^(-16*x - 8 
*x^2 + 2*x^3)*(-12 + 4*x))*Log[3 - x]^2),x]
 

Output:

((E^(2*x^3) + 2*E^(8*x*(2 + x)))*x)/(-(E^(8*x*(2 + x))*x) + (E^(2*x^3) + 2 
*E^(8*x*(2 + x)))*Log[3 - 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.

\(\displaystyle \int \frac {-e^{4 x^3-16 x^2-32 x} x+\left (e^{4 x^3-16 x^2-32 x} (x-3)+e^{2 x^3-8 x^2-16 x} (4 x-12)+4 x-12\right ) \log (3-x)+e^{2 x^3-8 x^2-16 x} \left (-6 x^5+34 x^4-32 x^3-48 x^2-4 x\right )-4 x}{x^3-3 x^2+\left (e^{4 x^3-16 x^2-32 x} (x-3)+e^{2 x^3-8 x^2-16 x} (4 x-12)+4 x-12\right ) \log ^2(3-x)+\left (-4 x^2+e^{2 x^3-8 x^2-16 x} \left (6 x-2 x^2\right )+12 x\right ) \log (3-x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (e^{4 x^3}+e^{2 x \left (x^2+4 x+8\right )} \left (6 x^4-34 x^3+32 x^2+48 x+4\right )+4 e^{16 x (x+2)}\right )-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 (x-3) \log (3-x)}{(3-x) \left (e^{8 x (x+2)} x-\left (e^{2 x^3}+2 e^{8 x (x+2)}\right ) \log (3-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (e^{2 x^3}+2 e^{8 x (x+2)}\right )^2 \log (3-x)}{\left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {4 e^{16 x (x+2)} x}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {e^{4 x^3} x}{(x-3) \left (e^{2 x^3} \log (3-x)-e^{8 x (x+2)} x+2 e^{8 x (x+2)} \log (3-x)\right )^2}-\frac {2 e^{2 x \left (x^2+4 x+8\right )} x \left (3 x^4-17 x^3+16 x^2+24 x+2\right )}{(x-3) \left (-e^{2 x^3} \log (3-x)+e^{8 x (x+2)} x-2 e^{8 x (x+2)} \log (3-x)\right )^2}\right )dx\)

Input:

Int[(-4*x - E^(-32*x - 16*x^2 + 4*x^3)*x + E^(-16*x - 8*x^2 + 2*x^3)*(-4*x 
 - 48*x^2 - 32*x^3 + 34*x^4 - 6*x^5) + (-12 + E^(-32*x - 16*x^2 + 4*x^3)*( 
-3 + x) + 4*x + E^(-16*x - 8*x^2 + 2*x^3)*(-12 + 4*x))*Log[3 - x])/(-3*x^2 
 + x^3 + (12*x - 4*x^2 + E^(-16*x - 8*x^2 + 2*x^3)*(6*x - 2*x^2))*Log[3 - 
x] + (-12 + E^(-32*x - 16*x^2 + 4*x^3)*(-3 + x) + 4*x + E^(-16*x - 8*x^2 + 
 2*x^3)*(-12 + 4*x))*Log[3 - x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 94.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43

method	result	size
risch	\(-\frac {x \left ({\mathrm e}^{2 x \left (x^{2}-4 x -8\right )}+2\right )}{-{\mathrm e}^{2 x \left (x^{2}-4 x -8\right )} \ln \left (-x +3\right )+x -2 \ln \left (-x +3\right )}\)	\(50\)
parallelrisch	\(\frac {-{\mathrm e}^{2 x^{3}-8 x^{2}-16 x} x -2 x}{-{\mathrm e}^{2 x^{3}-8 x^{2}-16 x} \ln \left (-x +3\right )+x -2 \ln \left (-x +3\right )}\)	\(59\)

Input:

int((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x-12)*ln 
(-x+3)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp(x^3-4* 
x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^ 
2+4*x-12)*ln(-x+3)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)*ln(-x+ 
3)+x^3-3*x^2),x,method=_RETURNVERBOSE)
 

Output:

-x*(exp(2*x*(x^2-4*x-8))+2)/(-exp(2*x*(x^2-4*x-8))*ln(-x+3)+x-2*ln(-x+3))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {x e^{\left (2 \, x^{3} - 8 \, x^{2} - 16 \, x\right )} + 2 \, x}{{\left (e^{\left (2 \, x^{3} - 8 \, x^{2} - 16 \, x\right )} + 2\right )} \log \left (-x + 3\right ) - x} \] Input:

integrate((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x- 
12)*log(3-x)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp( 
x^3-4*x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2 
-8*x)^2+4*x-12)*log(3-x)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)* 
log(3-x)+x^3-3*x^2),x, algorithm="fricas")
 

Output:

(x*e^(2*x^3 - 8*x^2 - 16*x) + 2*x)/((e^(2*x^3 - 8*x^2 - 16*x) + 2)*log(-x 
+ 3) - x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).

Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {x^{2}}{- x \log {\left (3 - x \right )} + e^{2 x^{3} - 8 x^{2} - 16 x} \log {\left (3 - x \right )}^{2} + 2 \log {\left (3 - x \right )}^{2}} + \frac {x}{\log {\left (3 - x \right )}} \] Input:

integrate((((-3+x)*exp(x**3-4*x**2-8*x)**4+(4*x-12)*exp(x**3-4*x**2-8*x)** 
2+4*x-12)*ln(3-x)-x*exp(x**3-4*x**2-8*x)**4+(-6*x**5+34*x**4-32*x**3-48*x* 
*2-4*x)*exp(x**3-4*x**2-8*x)**2-4*x)/(((-3+x)*exp(x**3-4*x**2-8*x)**4+(4*x 
-12)*exp(x**3-4*x**2-8*x)**2+4*x-12)*ln(3-x)**2+((-2*x**2+6*x)*exp(x**3-4* 
x**2-8*x)**2-4*x**2+12*x)*ln(3-x)+x**3-3*x**2),x)
 

Output:

x**2/(-x*log(3 - x) + exp(2*x**3 - 8*x**2 - 16*x)*log(3 - x)**2 + 2*log(3 
- x)**2) + x/log(3 - x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).

Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=-\frac {x e^{\left (2 \, x^{3}\right )} + 2 \, x e^{\left (8 \, x^{2} + 16 \, x\right )}}{x e^{\left (8 \, x^{2} + 16 \, x\right )} - {\left (e^{\left (2 \, x^{3}\right )} + 2 \, e^{\left (8 \, x^{2} + 16 \, x\right )}\right )} \log \left (-x + 3\right )} \] Input:

integrate((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x- 
12)*log(3-x)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp( 
x^3-4*x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2 
-8*x)^2+4*x-12)*log(3-x)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)* 
log(3-x)+x^3-3*x^2),x, algorithm="maxima")
 

Output:

-(x*e^(2*x^3) + 2*x*e^(8*x^2 + 16*x))/(x*e^(8*x^2 + 16*x) - (e^(2*x^3) + 2 
*e^(8*x^2 + 16*x))*log(-x + 3))
 

Giac [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {x e^{\left (2 \, x^{3} - 8 \, x^{2} - 16 \, x\right )} + 2 \, x}{e^{\left (2 \, x^{3} - 8 \, x^{2} - 16 \, x\right )} \log \left (-x + 3\right ) - x + 2 \, \log \left (-x + 3\right )} \] Input:

integrate((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x- 
12)*log(3-x)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp( 
x^3-4*x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2 
-8*x)^2+4*x-12)*log(3-x)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)* 
log(3-x)+x^3-3*x^2),x, algorithm="giac")
 

Output:

(x*e^(2*x^3 - 8*x^2 - 16*x) + 2*x)/(e^(2*x^3 - 8*x^2 - 16*x)*log(-x + 3) - 
 x + 2*log(-x + 3))
 

Mupad [B] (verification not implemented)

Time = 7.80 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.54 \[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\frac {6\,\ln \left (3-x\right )-x+3\,{\mathrm {e}}^{2\,x^3-8\,x^2-16\,x}\,\ln \left (3-x\right )+x\,{\mathrm {e}}^{2\,x^3-8\,x^2-16\,x}}{2\,\ln \left (3-x\right )-x+{\mathrm {e}}^{2\,x^3-8\,x^2-16\,x}\,\ln \left (3-x\right )} \] Input:

int(-(4*x - log(3 - x)*(4*x + exp(4*x^3 - 16*x^2 - 32*x)*(x - 3) + exp(2*x 
^3 - 8*x^2 - 16*x)*(4*x - 12) - 12) + exp(2*x^3 - 8*x^2 - 16*x)*(4*x + 48* 
x^2 + 32*x^3 - 34*x^4 + 6*x^5) + x*exp(4*x^3 - 16*x^2 - 32*x))/(log(3 - x) 
*(12*x + exp(2*x^3 - 8*x^2 - 16*x)*(6*x - 2*x^2) - 4*x^2) + log(3 - x)^2*( 
4*x + exp(4*x^3 - 16*x^2 - 32*x)*(x - 3) + exp(2*x^3 - 8*x^2 - 16*x)*(4*x 
- 12) - 12) - 3*x^2 + x^3),x)
 

Output:

(6*log(3 - x) - x + 3*exp(2*x^3 - 8*x^2 - 16*x)*log(3 - x) + x*exp(2*x^3 - 
 8*x^2 - 16*x))/(2*log(3 - x) - x + exp(2*x^3 - 8*x^2 - 16*x)*log(3 - x))
 

Reduce [F]

\[ \int \frac {-4 x-e^{-32 x-16 x^2+4 x^3} x+e^{-16 x-8 x^2+2 x^3} \left (-4 x-48 x^2-32 x^3+34 x^4-6 x^5\right )+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log (3-x)}{-3 x^2+x^3+\left (12 x-4 x^2+e^{-16 x-8 x^2+2 x^3} \left (6 x-2 x^2\right )\right ) \log (3-x)+\left (-12+e^{-32 x-16 x^2+4 x^3} (-3+x)+4 x+e^{-16 x-8 x^2+2 x^3} (-12+4 x)\right ) \log ^2(3-x)} \, dx=\int \frac {\left (\left (x -3\right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{4}+\left (4 x -12\right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{2}+4 x -12\right ) \mathrm {log}\left (-x +3\right )-x \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{4}+\left (-6 x^{5}+34 x^{4}-32 x^{3}-48 x^{2}-4 x \right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{2}-4 x}{\left (\left (x -3\right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{4}+\left (4 x -12\right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{2}+4 x -12\right ) \mathrm {log}\left (-x +3\right )^{2}+\left (\left (-2 x^{2}+6 x \right ) \left ({\mathrm e}^{x^{3}-4 x^{2}-8 x}\right )^{2}-4 x^{2}+12 x \right ) \mathrm {log}\left (-x +3\right )+x^{3}-3 x^{2}}d x \] Input:

int((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x-12)*lo 
g(3-x)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp(x^3-4* 
x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^ 
2+4*x-12)*log(3-x)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)*log(3- 
x)+x^3-3*x^2),x)
 

Output:

int((((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^2+4*x-12)*lo 
g(3-x)-x*exp(x^3-4*x^2-8*x)^4+(-6*x^5+34*x^4-32*x^3-48*x^2-4*x)*exp(x^3-4* 
x^2-8*x)^2-4*x)/(((-3+x)*exp(x^3-4*x^2-8*x)^4+(4*x-12)*exp(x^3-4*x^2-8*x)^ 
2+4*x-12)*log(3-x)^2+((-2*x^2+6*x)*exp(x^3-4*x^2-8*x)^2-4*x^2+12*x)*log(3- 
x)+x^3-3*x^2),x)