\(\int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} (262144+196608 x^2+e^{4 e^x} (4+3 x^2)+e^{2 e^x} (2048+1536 x^2+e^x (64 x+16 x^3)))}{e^{\frac {16}{256+e^{2 e^x}}} (327680+2560 e^{2 e^x}+5 e^{4 e^x})+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x)} \, dx\) [422]

Optimal result

Integrand size = 205, antiderivative size = 31 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \] Output:

1/5*(x^2+4)/(exp(4/(exp(exp(x))^2+256))^2-x)*x
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \] Input:

Integrate[(-131072*x^3 - 1024*E^(2*E^x)*x^3 - 2*E^(4*E^x)*x^3 + E^(8/(256 
+ E^(2*E^x)))*(262144 + 196608*x^2 + E^(4*E^x)*(4 + 3*x^2) + E^(2*E^x)*(20 
48 + 1536*x^2 + E^x*(64*x + 16*x^3))))/(E^(16/(256 + E^(2*E^x)))*(327680 + 
 2560*E^(2*E^x) + 5*E^(4*E^x)) + 327680*x^2 + 2560*E^(2*E^x)*x^2 + 5*E^(4* 
E^x)*x^2 + E^(8/(256 + E^(2*E^x)))*(-655360*x - 5120*E^(2*E^x)*x - 10*E^(4 
*E^x)*x)),x]
 

Output:

(x*(4 + x^2))/(5*(E^(8/(256 + E^(2*E^x))) - 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.

\(\displaystyle \int \frac {-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3-131072 x^3+e^{\frac {8}{e^{2 e^x}+256}} \left (196608 x^2+e^{4 e^x} \left (3 x^2+4\right )+e^{2 e^x} \left (e^x \left (16 x^3+64 x\right )+1536 x^2+2048\right )+262144\right )}{2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+327680 x^2+e^{\frac {16}{e^{2 e^x}+256}} \left (2560 e^{2 e^x}+5 e^{4 e^x}+327680\right )+e^{\frac {8}{e^{2 e^x}+256}} \left (-5120 e^{2 e^x} x-10 e^{4 e^x} x-655360 x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3-131072 x^3+e^{\frac {8}{e^{2 e^x}+256}} \left (196608 x^2+e^{4 e^x} \left (3 x^2+4\right )+e^{2 e^x} \left (e^x \left (16 x^3+64 x\right )+1536 x^2+2048\right )+262144\right )}{5 \left (e^{2 e^x}+256\right )^2 \left (e^{\frac {8}{e^{2 e^x}+256}}-x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int -\frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-e^{\frac {8}{256+e^{2 e^x}}} \left (196608 x^2+e^{4 e^x} \left (3 x^2+4\right )+16 e^{2 e^x} \left (96 x^2+e^x \left (x^3+4 x\right )+128\right )+262144\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-e^{\frac {8}{256+e^{2 e^x}}} \left (196608 x^2+e^{4 e^x} \left (3 x^2+4\right )+16 e^{2 e^x} \left (96 x^2+e^x \left (x^3+4 x\right )+128\right )+262144\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {196608 e^{\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {1536 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} x^2}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {262144 e^{\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {2048 e^{2 e^x+\frac {8}{256+e^{2 e^x}}}}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{5} \int \frac {1024 e^{2 e^x} x^3+2 e^{4 e^x} x^3+131072 x^3-16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x-65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )-e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\)

Input:

Int[(-131072*x^3 - 1024*E^(2*E^x)*x^3 - 2*E^(4*E^x)*x^3 + E^(8/(256 + E^(2 
*E^x)))*(262144 + 196608*x^2 + E^(4*E^x)*(4 + 3*x^2) + E^(2*E^x)*(2048 + 1 
536*x^2 + E^x*(64*x + 16*x^3))))/(E^(16/(256 + E^(2*E^x)))*(327680 + 2560* 
E^(2*E^x) + 5*E^(4*E^x)) + 327680*x^2 + 2560*E^(2*E^x)*x^2 + 5*E^(4*E^x)*x 
^2 + E^(8/(256 + E^(2*E^x)))*(-655360*x - 5120*E^(2*E^x)*x - 10*E^(4*E^x)* 
x)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 67.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {x \left (x^{2}+4\right )}{5 \left (-{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}+x \right )}\)	\(27\)
parallelrisch	\(-\frac {1024 x^{3}+4096 \,{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}}{5120 \left (-{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}+x \right )}\)	\(45\)

Input:

int((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp 
(x))^2+196608*x^2+262144)*exp(4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4 
-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x))^2+3 
27680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x) 
)^2-655360*x)*exp(4/(exp(exp(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*ex 
p(exp(x))^2+327680*x^2),x,method=_RETURNVERBOSE)
 

Output:

-1/5*x*(x^2+4)/(-exp(8/(exp(2*exp(x))+256))+x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \] Input:

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*e 
xp(exp(x))^2+196608*x^2+262144)*exp(4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp 
(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x 
))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp( 
exp(x))^2-655360*x)*exp(4/(exp(exp(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560* 
x^2*exp(exp(x))^2+327680*x^2),x, algorithm="fricas")
 

Output:

-1/5*(x^3 + 4*x)/(x - e^(8/(e^(2*e^x) + 256)))
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x^{3} + 4 x}{- 5 x + 5 e^{\frac {8}{e^{2 e^{x}} + 256}}} \] Input:

integrate((((3*x**2+4)*exp(exp(x))**4+((16*x**3+64*x)*exp(x)+1536*x**2+204 
8)*exp(exp(x))**2+196608*x**2+262144)*exp(4/(exp(exp(x))**2+256))**2-2*x** 
3*exp(exp(x))**4-1024*x**3*exp(exp(x))**2-131072*x**3)/((5*exp(exp(x))**4+ 
2560*exp(exp(x))**2+327680)*exp(4/(exp(exp(x))**2+256))**4+(-10*x*exp(exp( 
x))**4-5120*x*exp(exp(x))**2-655360*x)*exp(4/(exp(exp(x))**2+256))**2+5*x* 
*2*exp(exp(x))**4+2560*x**2*exp(exp(x))**2+327680*x**2),x)
 

Output:

(x**3 + 4*x)/(-5*x + 5*exp(8/(exp(2*exp(x)) + 256)))
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \] Input:

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*e 
xp(exp(x))^2+196608*x^2+262144)*exp(4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp 
(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x 
))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp( 
exp(x))^2-655360*x)*exp(4/(exp(exp(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560* 
x^2*exp(exp(x))^2+327680*x^2),x, algorithm="maxima")
 

Output:

-1/5*(x^3 + 4*x)/(x - e^(8/(e^(2*e^x) + 256)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (26) = 52\).

Time = 0.40 (sec) , antiderivative size = 566, normalized size of antiderivative = 18.26 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx =\text {Too large to display} \] Input:

integrate((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*e 
xp(exp(x))^2+196608*x^2+262144)*exp(4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp 
(x))^4-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x 
))^2+327680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp( 
exp(x))^2-655360*x)*exp(4/(exp(exp(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560* 
x^2*exp(exp(x))^2+327680*x^2),x, algorithm="giac")
 

Output:

-1/5*(16*x^4*e^(x + 7*e^x) + 8192*x^4*e^(x + 5*e^x) + 1048576*x^4*e^(x + 3 
*e^x) + x^3*e^(9*e^x) + 1024*x^3*e^(7*e^x) + 393216*x^3*e^(5*e^x) + 671088 
64*x^3*e^(3*e^x) + 4294967296*x^3*e^(e^x) + 64*x^2*e^(x + 7*e^x) + 32768*x 
^2*e^(x + 5*e^x) + 4194304*x^2*e^(x + 3*e^x) + 4*x*e^(9*e^x) + 4096*x*e^(7 
*e^x) + 1572864*x*e^(5*e^x) + 268435456*x*e^(3*e^x) + 17179869184*x*e^(e^x 
))/(16*x^2*e^(x + 7*e^x) + 8192*x^2*e^(x + 5*e^x) + 1048576*x^2*e^(x + 3*e 
^x) - 16*x*e^(x + 1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x 
) + 256) + 6*e^x + 1/32) - 8192*x*e^(x + 1/32*(32*e^(x + 2*e^x) + 8192*e^x 
 - e^(2*e^x))/(e^(2*e^x) + 256) + 4*e^x + 1/32) - 1048576*x*e^(x + 1/32*(3 
2*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 2*e^x + 1/32) 
+ x*e^(9*e^x) + 1024*x*e^(7*e^x) + 393216*x*e^(5*e^x) + 67108864*x*e^(3*e^ 
x) + 4294967296*x*e^(e^x) - e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^ 
x))/(e^(2*e^x) + 256) + 8*e^x + 1/32) - 1024*e^(1/32*(32*e^(x + 2*e^x) + 8 
192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 6*e^x + 1/32) - 393216*e^(1/32*(3 
2*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 256) + 4*e^x + 1/32) 
- 67108864*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e^(2*e^x))/(e^(2*e^x) + 
256) + 2*e^x + 1/32) - 4294967296*e^(1/32*(32*e^(x + 2*e^x) + 8192*e^x - e 
^(2*e^x))/(e^(2*e^x) + 256) + 1/32))
 

Mupad [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.42 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x\,{\left (512\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+{\mathrm {e}}^{4\,{\mathrm {e}}^x}+65536\right )}^2\,\left (2048\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+4\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+64\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+16\,x^3\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+65536\,x^2+512\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+262144\right )}{5\,{\left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}+256\right )}^2\,\left (x-{\mathrm {e}}^{\frac {8}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}+256}}\right )\,\left (67108864\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+393216\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+1024\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{8\,{\mathrm {e}}^x}+1048576\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+8192\,x\,{\mathrm {e}}^{x+4\,{\mathrm {e}}^x}+16\,x\,{\mathrm {e}}^{x+6\,{\mathrm {e}}^x}[phone]\right )} \] Input:

int(-(131072*x^3 + 1024*x^3*exp(2*exp(x)) + 2*x^3*exp(4*exp(x)) - exp(8/(e 
xp(2*exp(x)) + 256))*(exp(2*exp(x))*(exp(x)*(64*x + 16*x^3) + 1536*x^2 + 2 
048) + exp(4*exp(x))*(3*x^2 + 4) + 196608*x^2 + 262144))/(327680*x^2 - exp 
(8/(exp(2*exp(x)) + 256))*(655360*x + 5120*x*exp(2*exp(x)) + 10*x*exp(4*ex 
p(x))) + 2560*x^2*exp(2*exp(x)) + 5*x^2*exp(4*exp(x)) + exp(16/(exp(2*exp( 
x)) + 256))*(2560*exp(2*exp(x)) + 5*exp(4*exp(x)) + 327680)),x)
 

Output:

-(x*(512*exp(2*exp(x)) + exp(4*exp(x)) + 65536)^2*(2048*exp(2*exp(x)) + 4* 
exp(4*exp(x)) + 64*x*exp(x + 2*exp(x)) + 16*x^3*exp(x + 2*exp(x)) + 65536* 
x^2 + 512*x^2*exp(2*exp(x)) + x^2*exp(4*exp(x)) + 262144))/(5*(exp(2*exp(x 
)) + 256)^2*(x - exp(8/(exp(2*exp(x)) + 256)))*(67108864*exp(2*exp(x)) + 3 
93216*exp(4*exp(x)) + 1024*exp(6*exp(x)) + exp(8*exp(x)) + 1048576*x*exp(x 
 + 2*exp(x)) + 8192*x*exp(x + 4*exp(x)) + 16*x*exp(x + 6*exp(x)) + 4294967 
296))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {4 e^{\frac {8}{e^{2 e^{x}}+256}}+x^{3}}{5 e^{\frac {8}{e^{2 e^{x}}+256}}-5 x} \] Input:

int((((3*x^2+4)*exp(exp(x))^4+((16*x^3+64*x)*exp(x)+1536*x^2+2048)*exp(exp 
(x))^2+196608*x^2+262144)*exp(4/(exp(exp(x))^2+256))^2-2*x^3*exp(exp(x))^4 
-1024*x^3*exp(exp(x))^2-131072*x^3)/((5*exp(exp(x))^4+2560*exp(exp(x))^2+3 
27680)*exp(4/(exp(exp(x))^2+256))^4+(-10*x*exp(exp(x))^4-5120*x*exp(exp(x) 
)^2-655360*x)*exp(4/(exp(exp(x))^2+256))^2+5*x^2*exp(exp(x))^4+2560*x^2*ex 
p(exp(x))^2+327680*x^2),x)
 

Output:

(4*e**(8/(e**(2*e**x) + 256)) + x**3)/(5*(e**(8/(e**(2*e**x) + 256)) - x))