Integrand size = 205, antiderivative size = 31 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \]
Time = 0.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x \left (4+x^2\right )}{5 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )} \]
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]
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 |
\(\displaystyle -\frac {1}{5} \int \left (\frac {1024 e^{2 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {2 e^{4 e^x} x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}+\frac {131072 x^3}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {16 e^{x+2 e^x+\frac {8}{256+e^{2 e^x}}} \left (x^2+4\right ) x}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {65536 e^{\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {512 e^{2 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}-\frac {e^{4 e^x+\frac {8}{256+e^{2 e^x}}} \left (3 x^2+4\right )}{\left (256+e^{2 e^x}\right )^2 \left (e^{\frac {8}{256+e^{2 e^x}}}-x\right )^2}\right )dx\) |
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]
3.5.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 56.90 (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 | \(-\frac {1024 x^{3}+4096 \,{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}}{5120 \left (-{\mathrm e}^{\frac {8}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+256}}+x \right )}\) | \(45\) |
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)
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \]
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=\
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=\frac {x^{3} + 4 x}{- 5 x + 5 e^{\frac {8}{e^{2 e^{x}} + 256}}} \]
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)
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x^{3} + 4 \, x}{5 \, {\left (x - e^{\left (\frac {8}{e^{\left (2 \, e^{x}\right )} + 256}\right )}\right )}} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (26) = 52\).
Time = 0.57 (sec) , antiderivative size = 566, normalized size of antiderivative = 18.26 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx =\text {Too large to display} \]
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=\
-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))
Time = 7.97 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.42 \[ \int \frac {-131072 x^3-1024 e^{2 e^x} x^3-2 e^{4 e^x} x^3+e^{\frac {8}{256+e^{2 e^x}}} \left (262144+196608 x^2+e^{4 e^x} \left (4+3 x^2\right )+e^{2 e^x} \left (2048+1536 x^2+e^x \left (64 x+16 x^3\right )\right )\right )}{e^{\frac {16}{256+e^{2 e^x}}} \left (327680+2560 e^{2 e^x}+5 e^{4 e^x}\right )+327680 x^2+2560 e^{2 e^x} x^2+5 e^{4 e^x} x^2+e^{\frac {8}{256+e^{2 e^x}}} \left (-655360 x-5120 e^{2 e^x} x-10 e^{4 e^x} x\right )} \, dx=-\frac {x\,{\left (512\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+{\mathrm {e}}^{4\,{\mathrm {e}}^x}+65536\right )}^2\,\left (2048\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+4\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+64\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+16\,x^3\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+65536\,x^2+512\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+262144\right )}{5\,{\left ({\mathrm {e}}^{2\,{\mathrm {e}}^x}+256\right )}^2\,\left (x-{\mathrm {e}}^{\frac {8}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}+256}}\right )\,\left (67108864\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+393216\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}+1024\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{8\,{\mathrm {e}}^x}+1048576\,x\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^x}+8192\,x\,{\mathrm {e}}^{x+4\,{\mathrm {e}}^x}+16\,x\,{\mathrm {e}}^{x+6\,{\mathrm {e}}^x}+4294967296\right )} \]
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)
-(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))