\(\int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} (2 e^{5+x} x-4 e^5 x^2)+e^x (-2 x^2+4 x^3))}{e^{2 x}-2 e^x x^2+x^4} \, dx\) [155]

Optimal result

Integrand size = 142, antiderivative size = 27 \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=e^{e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=e^{e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \] Input:

Integrate[(E^(-2*E^(E^5/(E^x - x^2)) - 2*x + E^(-2*E^(E^5/(E^x - x^2)) - 2 
*x)*x)*(E^(2*x)*(1 - 2*x) + x^4 - 2*x^5 + E^(E^5/(E^x - x^2))*(2*E^(5 + x) 
*x - 4*E^5*x^2) + E^x*(-2*x^2 + 4*x^3)))/(E^(2*x) - 2*E^x*x^2 + x^4),x]
 

Output:

E^(E^(-2*E^(E^5/(E^x - x^2)) - 2*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.

\(\displaystyle \int \frac {\left (-2 x^5+x^4+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{x+5} x-4 e^5 x^2\right )+e^x \left (4 x^3-2 x^2\right )+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{x^4-2 e^x x^2+e^{2 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-2 x^5+x^4+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{x+5} x-4 e^5 x^2\right )+e^x \left (4 x^3-2 x^2\right )+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (-2 x^5+x^4-4 e^{\frac {e^5}{e^x-x^2}+5} x^2+2 e^x (2 x-1) x^2+2 e^{\frac {e^5}{e^x-x^2}+x+5} x+e^{2 x} (1-2 x)\right ) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (e^x-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (2 x-1) x^2 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-x\right )}{\left (e^x-x^2\right )^2}+\frac {2 \left (e^x-2 x\right ) x \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}+\frac {e^5}{e^x-x^2}-2 x+5\right )}{\left (e^x-x^2\right )^2}-\frac {(2 x-1) \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}\right )}{\left (e^x-x^2\right )^2}-\frac {2 x^5 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}+\frac {x^4 \exp \left (e^{-2 \left (e^{\frac {e^5}{e^x-x^2}}+x\right )} x-2 e^{\frac {e^5}{e^x-x^2}}-2 x\right )}{\left (x^2-e^x\right )^2}\right )dx\)

Input:

Int[(E^(-2*E^(E^5/(E^x - x^2)) - 2*x + E^(-2*E^(E^5/(E^x - x^2)) - 2*x)*x) 
*(E^(2*x)*(1 - 2*x) + x^4 - 2*x^5 + E^(E^5/(E^x - x^2))*(2*E^(5 + x)*x - 4 
*E^5*x^2) + E^x*(-2*x^2 + 4*x^3)))/(E^(2*x) - 2*E^x*x^2 + x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[{\mathrm e}^{x \,{\mathrm e}^{-2 \,{\mathrm e}^{\frac {{\mathrm e}^{5}}{{\mathrm e}^{x}-x^{2}}}-2 x}}\]

Input:

int(((2*x*exp(5)*exp(x)-4*x^2*exp(5))*exp(exp(5)/(exp(x)-x^2))+(1-2*x)*exp 
(x)^2+(4*x^3-2*x^2)*exp(x)-2*x^5+x^4)*exp(x/exp(2*exp(exp(5)/(exp(x)-x^2)) 
+2*x))/(exp(x)^2-2*exp(x)*x^2+x^4)/exp(2*exp(exp(5)/(exp(x)-x^2))+2*x),x)
 

Output:

exp(x*exp(-2*exp(exp(5)/(exp(x)-x^2))-2*x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=e^{\left (x e^{\left (-2 \, x - 2 \, e^{\left (-\frac {e^{10}}{x^{2} e^{5} - e^{\left (x + 5\right )}}\right )}\right )}\right )} \] Input:

integrate(((2*x*exp(5)*exp(x)-4*x^2*exp(5))*exp(exp(5)/(exp(x)-x^2))+(1-2* 
x)*exp(x)^2+(4*x^3-2*x^2)*exp(x)-2*x^5+x^4)*exp(x/exp(2*exp(exp(5)/(exp(x) 
-x^2))+2*x))/(exp(x)^2-2*exp(x)*x^2+x^4)/exp(2*exp(exp(5)/(exp(x)-x^2))+2* 
x),x, algorithm="fricas")
 

Output:

e^(x*e^(-2*x - 2*e^(-e^10/(x^2*e^5 - e^(x + 5)))))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=\text {Timed out} \] Input:

integrate(((2*x*exp(5)*exp(x)-4*x**2*exp(5))*exp(exp(5)/(exp(x)-x**2))+(1- 
2*x)*exp(x)**2+(4*x**3-2*x**2)*exp(x)-2*x**5+x**4)*exp(x/exp(2*exp(exp(5)/ 
(exp(x)-x**2))+2*x))/(exp(x)**2-2*exp(x)*x**2+x**4)/exp(2*exp(exp(5)/(exp( 
x)-x**2))+2*x),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=e^{\left (x e^{\left (-2 \, x - 2 \, e^{\left (-\frac {e^{5}}{x^{2} - e^{x}}\right )}\right )}\right )} \] Input:

integrate(((2*x*exp(5)*exp(x)-4*x^2*exp(5))*exp(exp(5)/(exp(x)-x^2))+(1-2* 
x)*exp(x)^2+(4*x^3-2*x^2)*exp(x)-2*x^5+x^4)*exp(x/exp(2*exp(exp(5)/(exp(x) 
-x^2))+2*x))/(exp(x)^2-2*exp(x)*x^2+x^4)/exp(2*exp(exp(5)/(exp(x)-x^2))+2* 
x),x, algorithm="maxima")
 

Output:

e^(x*e^(-2*x - 2*e^(-e^5/(x^2 - e^x))))
 

Giac [F]

\[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=\int { -\frac {{\left (2 \, x^{5} - x^{4} + {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{3} - x^{2}\right )} e^{x} + 2 \, {\left (2 \, x^{2} e^{5} - x e^{\left (x + 5\right )}\right )} e^{\left (-\frac {e^{5}}{x^{2} - e^{x}}\right )}\right )} e^{\left (x e^{\left (-2 \, x - 2 \, e^{\left (-\frac {e^{5}}{x^{2} - e^{x}}\right )}\right )} - 2 \, x - 2 \, e^{\left (-\frac {e^{5}}{x^{2} - e^{x}}\right )}\right )}}{x^{4} - 2 \, x^{2} e^{x} + e^{\left (2 \, x\right )}} \,d x } \] Input:

integrate(((2*x*exp(5)*exp(x)-4*x^2*exp(5))*exp(exp(5)/(exp(x)-x^2))+(1-2* 
x)*exp(x)^2+(4*x^3-2*x^2)*exp(x)-2*x^5+x^4)*exp(x/exp(2*exp(exp(5)/(exp(x) 
-x^2))+2*x))/(exp(x)^2-2*exp(x)*x^2+x^4)/exp(2*exp(exp(5)/(exp(x)-x^2))+2* 
x),x, algorithm="giac")
 

Output:

integrate(-(2*x^5 - x^4 + (2*x - 1)*e^(2*x) - 2*(2*x^3 - x^2)*e^x + 2*(2*x 
^2*e^5 - x*e^(x + 5))*e^(-e^5/(x^2 - e^x)))*e^(x*e^(-2*x - 2*e^(-e^5/(x^2 
- e^x))) - 2*x - 2*e^(-e^5/(x^2 - e^x)))/(x^4 - 2*x^2*e^x + e^(2*x)), x)
 

Mupad [B] (verification not implemented)

Time = 4.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{{\mathrm {e}}^x-x^2}}}} \] Input:

int(-(exp(x*exp(- 2*x - 2*exp(exp(5)/(exp(x) - x^2))))*exp(- 2*x - 2*exp(e 
xp(5)/(exp(x) - x^2)))*(exp(x)*(2*x^2 - 4*x^3) + exp(exp(5)/(exp(x) - x^2) 
)*(4*x^2*exp(5) - 2*x*exp(5)*exp(x)) + exp(2*x)*(2*x - 1) - x^4 + 2*x^5))/ 
(exp(2*x) - 2*x^2*exp(x) + x^4),x)
 

Output:

exp(x*exp(-2*x)*exp(-2*exp(exp(5)/(exp(x) - x^2))))
 

Reduce [F]

\[ \int \frac {e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x+e^{-2 e^{\frac {e^5}{e^x-x^2}}-2 x} x} \left (e^{2 x} (1-2 x)+x^4-2 x^5+e^{\frac {e^5}{e^x-x^2}} \left (2 e^{5+x} x-4 e^5 x^2\right )+e^x \left (-2 x^2+4 x^3\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx=\text {too large to display} \] Input:

int(((2*x*exp(5)*exp(x)-4*x^2*exp(5))*exp(exp(5)/(exp(x)-x^2))+(1-2*x)*exp 
(x)^2+(4*x^3-2*x^2)*exp(x)-2*x^5+x^4)*exp(x/exp(2*exp(exp(5)/(exp(x)-x^2)) 
+2*x))/(exp(x)^2-2*exp(x)*x^2+x^4)/exp(2*exp(exp(5)/(exp(x)-x^2))+2*x),x)
 

Output:

int(e**(x/e**(2*e**(e**5/(e**x - x**2)) + 2*x))/(e**(2*e**(e**5/(e**x - x* 
*2)) + 2*x) - 2*e**(2*e**(e**5/(e**x - x**2)) + x)*x**2 + e**(2*e**(e**5/( 
e**x - x**2)))*x**4),x) - 4*int((e**((e**(2*e**(e**5/(e**x - x**2)) + x)*e 
**5 + x)/(e**(2*e**(e**5/(e**x - x**2)) + 2*x) - e**(2*e**(e**5/(e**x - x* 
*2)) + x)*x**2))*x**2)/(e**((2*e**((2*e**((e**x*x + e**5 - x**3)/(e**x - x 
**2)) - 2*e**(e**5/(e**x - x**2))*x**2 + 3*e**x*x + e**5 - 3*x**3)/(e**x - 
 x**2)) - 2*e**((2*e**((e**x*x + e**5 - x**3)/(e**x - x**2)) - 2*e**(e**5/ 
(e**x - x**2))*x**2 + 2*e**x*x + e**5 - 2*x**3)/(e**x - x**2))*x**2 + 4*e* 
*(2*e**(e**5/(e**x - x**2)) + 3*x)*x - 4*e**(2*e**(e**5/(e**x - x**2)) + 2 
*x)*x**3 + x**3)/(e**(2*e**(e**5/(e**x - x**2)) + 3*x) - e**(2*e**(e**5/(e 
**x - x**2)) + 2*x)*x**2)) - 2*e**((2*e**((2*e**((e**x*x + e**5 - x**3)/(e 
**x - x**2)) - 2*e**(e**5/(e**x - x**2))*x**2 + 3*e**x*x + e**5 - 3*x**3)/ 
(e**x - x**2)) - 2*e**((2*e**((e**x*x + e**5 - x**3)/(e**x - x**2)) - 2*e* 
*(e**5/(e**x - x**2))*x**2 + 2*e**x*x + e**5 - 2*x**3)/(e**x - x**2))*x**2 
 + 3*e**(2*e**(e**5/(e**x - x**2)) + 3*x)*x - 3*e**(2*e**(e**5/(e**x - x** 
2)) + 2*x)*x**3 + x**3)/(e**(2*e**(e**5/(e**x - x**2)) + 3*x) - e**(2*e**( 
e**5/(e**x - x**2)) + 2*x)*x**2))*x**2 + e**((2*e**((2*e**((e**x*x + e**5 
- x**3)/(e**x - x**2)) - 2*e**(e**5/(e**x - x**2))*x**2 + 3*e**x*x + e**5 
- 3*x**3)/(e**x - x**2)) - 2*e**((2*e**((e**x*x + e**5 - x**3)/(e**x - x** 
2)) - 2*e**(e**5/(e**x - x**2))*x**2 + 2*e**x*x + e**5 - 2*x**3)/(e**x ...