\(\int \frac {e^{\frac {1}{4} (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 (-4-4 x-x^2))} (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} (2+2 e^x)+e^{2+2 x} (-50-25 x)+e^{2+x} (20-90 x-50 x^2)+e^2 (-2+19 x-40 x^2-25 x^3))}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx\) [419]

Optimal result

Integrand size = 159, antiderivative size = 32 \[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=e^{e^{3+\frac {1}{-5+\frac {1}{e^x+x}}}-e^2 \left (1+\frac {x}{2}\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=e^{\frac {1}{4} e^2 \left (4 e^{\frac {4}{5}+\frac {1}{5-25 e^x-25 x}}-(2+x)^2\right )} \] Input:

Integrate[(E^((4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4 
*x - x^2))/4)*(E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x))*(2 + 2*E^x) + E 
^(2 + 2*x)*(-50 - 25*x) + E^(2 + x)*(20 - 90*x - 50*x^2) + E^2*(-2 + 19*x 
- 40*x^2 - 25*x^3)))/(2 + 50*E^(2*x) - 20*x + 50*x^2 + E^x*(-20 + 100*x)), 
x]
 

Output:

E^((E^2*(4*E^(4/5 + (5 - 25*E^x - 25*x)^(-1)) - (2 + x)^2))/4)
 

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.

\(\displaystyle \int \frac {\left (e^{x+2} \left (-50 x^2-90 x+20\right )+e^2 \left (-25 x^3-40 x^2+19 x-2\right )+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (2 e^x+2\right )+e^{2 x+2} (-25 x-50)\right ) \exp \left (\frac {1}{4} \left (e^2 \left (-x^2-4 x-4\right )+4 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}\right )\right )}{50 x^2-20 x+50 e^{2 x}+e^x (100 x-20)+2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (e^{x+2} \left (-50 x^2-90 x+20\right )+e^2 \left (-25 x^3-40 x^2+19 x-2\right )+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (2 e^x+2\right )+e^{2 x+2} (-25 x-50)\right ) \exp \left (\frac {1}{4} \left (e^2 \left (-x^2-4 x-4\right )+4 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}\right )\right )}{2 \left (-5 x-5 e^x+1\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {1}{4} \left (4 e^{\frac {-14 x-14 e^x+3}{-5 x-5 e^x+1}}-e^2 \left (x^2+4 x+4\right )\right )\right ) \left (2 e^{\frac {-14 x-14 e^x+3}{-5 x-5 e^x+1}} \left (1+e^x\right )-25 e^{2 x+2} (x+2)+10 e^{x+2} \left (-5 x^2-9 x+2\right )-e^2 \left (25 x^3+40 x^2-19 x+2\right )\right )}{\left (-5 x-5 e^x+1\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (\frac {14 x+14 e^x-3}{5 x+5 e^x-1}+\frac {1}{4} \left (4 e^{\frac {-14 x-14 e^x+3}{-5 x-5 e^x+1}}-e^2 \left (x^2+4 x+4\right )\right )\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (\frac {1}{4} \left (4 e^{\frac {-14 x-14 e^x+3}{-5 x-5 e^x+1}}-e^2 \left (x^2+4 x+4\right )\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+\frac {14 e^x}{5 x+5 e^x-1}+\frac {14 x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+\exp \left (\frac {14 x}{5 x+5 e^x-1}+\frac {14 e^x}{5 x+5 e^x-1}-\frac {3}{5 x+5 e^x-1}\right )+2\right ) (x+2)\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {1}{2} \int \exp \left (e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}-\frac {1}{4} e^2 (x+2)^2\right ) \left (\frac {2 e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}} \left (1+e^x\right )}{\left (5 x+5 e^x-1\right )^2}-e^2 (x+2)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+\frac {14 x+14 e^x-3}{5 x+5 e^x-1}\right ) \left (1+e^x\right )}{\left (-5 x-5 e^x+1\right )^2}-\exp \left (-\frac {1}{4} e^2 (x+2)^2+e^{\frac {14 x+14 e^x-3}{5 x+5 e^x-1}}+2\right ) (x+2)\right )dx\)

Input:

Int[(E^((4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x 
^2))/4)*(E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x))*(2 + 2*E^x) + E^(2 + 
2*x)*(-50 - 25*x) + E^(2 + x)*(20 - 90*x - 50*x^2) + E^2*(-2 + 19*x - 40*x 
^2 - 25*x^3)))/(2 + 50*E^(2*x) - 20*x + 50*x^2 + E^x*(-20 + 100*x)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {14 \,{\mathrm e}^{x}+14 x -3}{5 \,{\mathrm e}^{x}+5 x -1}}-\frac {x^{2} {\mathrm e}^{2}}{4}-{\mathrm e}^{2} x -{\mathrm e}^{2}}\)	\(41\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{\frac {14 \,{\mathrm e}^{x}+14 x -3}{5 \,{\mathrm e}^{x}+5 x -1}}+\frac {\left (-x^{2}-4 x -4\right ) {\mathrm e}^{2}}{4}}\)	\(41\)

Input:

int(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp( 
1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*ex 
p(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4)*exp( 
1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x,method=_RETURNVERBOS 
E)
 

Output:

exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))-1/4*x^2*exp(2)-exp(2)*x-exp(2 
))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {{\left (14 \, x - 3\right )} e^{2} + 14 \, e^{\left (x + 2\right )}}{{\left (5 \, x - 1\right )} e^{2} + 5 \, e^{\left (x + 2\right )}}\right )}\right )} \] Input:

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50 
)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x 
-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4 
)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="fr 
icas")
 

Output:

e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^(((14*x - 3)*e^2 + 14*e^(x + 2))/((5*x - 1 
)*e^2 + 5*e^(x + 2))))
 

Sympy [A] (verification not implemented)

Time = 1.91 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=e^{\left (- \frac {x^{2}}{4} - x - 1\right ) e^{2} + e^{\frac {14 x + 14 e^{x} - 3}{5 x + 5 e^{x} - 1}}} \] Input:

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50 
)*exp(1)**2*exp(x)**2+(-50*x**2-90*x+20)*exp(1)**2*exp(x)+(-25*x**3-40*x** 
2+19*x-2)*exp(1)**2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x* 
*2-4*x-4)*exp(1)**2)/(50*exp(x)**2+(100*x-20)*exp(x)+50*x**2-20*x+2),x)
 

Output:

exp((-x**2/4 - x - 1)*exp(2) + exp((14*x + 14*exp(x) - 3)/(5*x + 5*exp(x) 
- 1)))
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=\int { -\frac {{\left ({\left (25 \, x^{3} + 40 \, x^{2} - 19 \, x + 2\right )} e^{2} + 25 \, {\left (x + 2\right )} e^{\left (2 \, x + 2\right )} + 10 \, {\left (5 \, x^{2} + 9 \, x - 2\right )} e^{\left (x + 2\right )} - 2 \, {\left (e^{x} + 1\right )} e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )} e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )}}{2 \, {\left (25 \, x^{2} + 10 \, {\left (5 \, x - 1\right )} e^{x} - 10 \, x + 25 \, e^{\left (2 \, x\right )} + 1\right )}} \,d x } \] Input:

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50 
)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x 
-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4 
)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="ma 
xima")
 

Output:

-1/2*integrate(((25*x^3 + 40*x^2 - 19*x + 2)*e^2 + 25*(x + 2)*e^(2*x + 2) 
+ 10*(5*x^2 + 9*x - 2)*e^(x + 2) - 2*(e^x + 1)*e^((14*x + 14*e^x - 3)/(5*x 
 + 5*e^x - 1)))*e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^((14*x + 14*e^x - 3)/(5*x 
+ 5*e^x - 1)))/(25*x^2 + 10*(5*x - 1)*e^x - 10*x + 25*e^(2*x) + 1), x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=\int { -\frac {{\left ({\left (25 \, x^{3} + 40 \, x^{2} - 19 \, x + 2\right )} e^{2} + 25 \, {\left (x + 2\right )} e^{\left (2 \, x + 2\right )} + 10 \, {\left (5 \, x^{2} + 9 \, x - 2\right )} e^{\left (x + 2\right )} - 2 \, {\left (e^{x} + 1\right )} e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )} e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )}}{2 \, {\left (25 \, x^{2} + 10 \, {\left (5 \, x - 1\right )} e^{x} - 10 \, x + 25 \, e^{\left (2 \, x\right )} + 1\right )}} \,d x } \] Input:

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50 
)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x 
-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4 
)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="gi 
ac")
 

Output:

integrate(-1/2*((25*x^3 + 40*x^2 - 19*x + 2)*e^2 + 25*(x + 2)*e^(2*x + 2) 
+ 10*(5*x^2 + 9*x - 2)*e^(x + 2) - 2*(e^x + 1)*e^((14*x + 14*e^x - 3)/(5*x 
 + 5*e^x - 1)))*e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^((14*x + 14*e^x - 3)/(5*x 
+ 5*e^x - 1)))/(25*x^2 + 10*(5*x - 1)*e^x - 10*x + 25*e^(2*x) + 1), x)
 

Mupad [B] (verification not implemented)

Time = 3.42 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx={\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^2}{4}}\,{\mathrm {e}}^{-{\mathrm {e}}^2}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^2}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {14\,x}{5\,x+5\,{\mathrm {e}}^x-1}}\,{\mathrm {e}}^{\frac {14\,{\mathrm {e}}^x}{5\,x+5\,{\mathrm {e}}^x-1}}\,{\mathrm {e}}^{-\frac {3}{5\,x+5\,{\mathrm {e}}^x-1}}} \] Input:

int(-(exp(exp((14*x + 14*exp(x) - 3)/(5*x + 5*exp(x) - 1)) - (exp(2)*(4*x 
+ x^2 + 4))/4)*(exp(2)*(40*x^2 - 19*x + 25*x^3 + 2) - exp((14*x + 14*exp(x 
) - 3)/(5*x + 5*exp(x) - 1))*(2*exp(x) + 2) + exp(2)*exp(x)*(90*x + 50*x^2 
 - 20) + exp(2*x)*exp(2)*(25*x + 50)))/(50*exp(2*x) - 20*x + exp(x)*(100*x 
 - 20) + 50*x^2 + 2),x)
 

Output:

exp(-(x^2*exp(2))/4)*exp(-exp(2))*exp(-x*exp(2))*exp(exp((14*x)/(5*x + 5*e 
xp(x) - 1))*exp((14*exp(x))/(5*x + 5*exp(x) - 1))*exp(-3/(5*x + 5*exp(x) - 
 1)))
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )} \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx=\int \frac {\left (\left (2 \,{\mathrm e}^{x}+2\right ) {\mathrm e}^{\frac {14 \,{\mathrm e}^{x}+14 x -3}{5 \,{\mathrm e}^{x}+5 x -1}}+\left (-25 x -50\right ) \left ({\mathrm e}\right )^{2} \left ({\mathrm e}^{x}\right )^{2}+\left (-50 x^{2}-90 x +20\right ) \left ({\mathrm e}\right )^{2} {\mathrm e}^{x}+\left (-25 x^{3}-40 x^{2}+19 x -2\right ) \left ({\mathrm e}\right )^{2}\right ) {\mathrm e}^{{\mathrm e}^{\frac {14 \,{\mathrm e}^{x}+14 x -3}{5 \,{\mathrm e}^{x}+5 x -1}}+\frac {\left (-x^{2}-4 x -4\right ) \left ({\mathrm e}\right )^{2}}{4}}}{50 \left ({\mathrm e}^{x}\right )^{2}+\left (100 x -20\right ) {\mathrm e}^{x}+50 x^{2}-20 x +2}d x \] Input:

int(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp( 
1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*ex 
p(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4)*exp( 
1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x)
 

Output:

int(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp( 
1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*ex 
p(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4)*exp( 
1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x)