Integrand size = 290, antiderivative size = 28 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=1+\frac {1}{6-\left (e^{\frac {5 x}{-e^5+2 x}}+x\right )^2} \] Output:
1/(6-(x+exp(5/(-exp(5)+2*x)*x))^2)+1
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=-\frac {1}{-6+e^{-\frac {10 x}{e^5-2 x}}+2 e^{-\frac {5 x}{e^5-2 x}} x+x^2} \] Input:
Integrate[(-10*E^(5 - (10*x)/(E^5 - 2*x)) + 2*E^10*x - 8*E^5*x^2 + 8*x^3 + (2*E^10 - 18*E^5*x + 8*x^2)/E^((5*x)/(E^5 - 2*x)))/(144*x^2 - 48*x^4 + 4* x^6 + (E^10 - 4*E^5*x + 4*x^2)/E^((20*x)/(E^5 - 2*x)) + (4*E^10*x - 16*E^5 *x^2 + 16*x^3)/E^((15*x)/(E^5 - 2*x)) + E^10*(36 - 12*x^2 + x^4) + E^5*(-1 44*x + 48*x^3 - 4*x^5) + (-48*x^2 + 24*x^4 + E^10*(-12 + 6*x^2) + E^5*(48* x - 24*x^3))/E^((10*x)/(E^5 - 2*x)) + (-96*x^3 + 16*x^5 + E^10*(-24*x + 4* x^3) + E^5*(96*x^2 - 16*x^4))/E^((5*x)/(E^5 - 2*x))),x]
Output:
-(-6 + E^((-10*x)/(E^5 - 2*x)) + (2*x)/E^((5*x)/(E^5 - 2*x)) + x^2)^(-1)
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.
| \(\displaystyle \int \frac {8 x^3-8 e^5 x^2+e^{-\frac {5 x}{e^5-2 x}} \left (8 x^2-18 e^5 x+2 e^{10}\right )+2 e^{10} x-10 e^{5-\frac {10 x}{e^5-2 x}}}{4 x^6-48 x^4+144 x^2+e^{-\frac {20 x}{e^5-2 x}} \left (4 x^2-4 e^5 x+e^{10}\right )+e^5 \left (-4 x^5+48 x^3-144 x\right )+e^{10} \left (x^4-12 x^2+36\right )+e^{-\frac {15 x}{e^5-2 x}} \left (16 x^3-16 e^5 x^2+4 e^{10} x\right )+e^{-\frac {10 x}{e^5-2 x}} \left (24 x^4+e^5 \left (48 x-24 x^3\right )-48 x^2+e^{10} \left (6 x^2-12\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (16 x^5-96 x^3+e^{10} \left (4 x^3-24 x\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (4 e^{\frac {5 x}{e^5-2 x}} x^2-4 e^{\frac {5 x}{e^5-2 x}+5} x+e^{\frac {5 x}{e^5-2 x}+10}-5 e^5\right )}{\left (e^5-2 x\right )^2 \left (e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+2 e^{\frac {5 x}{e^5-2 x}} x+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x-e^{\frac {10 x}{e^5-2 x}} \left (6-x^2\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x-e^{\frac {10 x}{e^5-2 x}} \left (6-x^2\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {e^{\frac {10 x}{e^5-2 x}} \left (e^{\frac {5 x}{e^5-2 x}} x+1\right ) \left (-4 e^{\frac {5 x}{e^5-2 x}} x^2+4 e^{\frac {5 x}{e^5-2 x}+5} x-e^{\frac {5 x}{e^5-2 x}+10}+5 e^5\right )}{\left (e^5-2 x\right )^2 \left (2 e^{\frac {5 x}{e^5-2 x}} x+e^{\frac {10 x}{e^5-2 x}} \left (x^2-6\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {e^{\frac {10 x}{e^5-2 x}} \left (-4 e^{\frac {5 x}{e^5-2 x}} x^4-e^{\frac {5 x}{e^5-2 x}+5} x^3-4 x^3-24 e^{\frac {5 x}{e^5-2 x}} \left (1+\frac {e^{10}}{24}\right ) x^2-e^5 x^2+54 e^{\frac {5 x}{e^5-2 x}+5} x-e^{10} x-6 e^{\frac {5 x}{e^5-2 x}+10}+30 e^5\right )}{\left (e^5-2 x\right )^2 \left (6-x^2\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )^2}-\frac {e^{\frac {10 x}{e^5-2 x}} x}{\left (x^2-6\right ) \left (e^{\frac {10 x}{e^5-2 x}} x^2+2 e^{\frac {5 x}{e^5-2 x}} x-6 e^{\frac {10 x}{e^5-2 x}}+1\right )}\right )dx\) |
Input:
Int[(-10*E^(5 - (10*x)/(E^5 - 2*x)) + 2*E^10*x - 8*E^5*x^2 + 8*x^3 + (2*E^ 10 - 18*E^5*x + 8*x^2)/E^((5*x)/(E^5 - 2*x)))/(144*x^2 - 48*x^4 + 4*x^6 + (E^10 - 4*E^5*x + 4*x^2)/E^((20*x)/(E^5 - 2*x)) + (4*E^10*x - 16*E^5*x^2 + 16*x^3)/E^((15*x)/(E^5 - 2*x)) + E^10*(36 - 12*x^2 + x^4) + E^5*(-144*x + 48*x^3 - 4*x^5) + (-48*x^2 + 24*x^4 + E^10*(-12 + 6*x^2) + E^5*(48*x - 24 *x^3))/E^((10*x)/(E^5 - 2*x)) + (-96*x^3 + 16*x^5 + E^10*(-24*x + 4*x^3) + E^5*(96*x^2 - 16*x^4))/E^((5*x)/(E^5 - 2*x))),x]
Output:
$Aborted
Time = 3.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {1}{{\mathrm e}^{-\frac {10 x}{{\mathrm e}^{5}-2 x}}+2 \,{\mathrm e}^{-\frac {5 x}{{\mathrm e}^{5}-2 x}} x +x^{2}-6}\) | \(37\) |
| parallelrisch | \(-\frac {1}{{\mathrm e}^{-\frac {10 x}{{\mathrm e}^{5}-2 x}}+2 \,{\mathrm e}^{-\frac {5 x}{{\mathrm e}^{5}-2 x}} x +x^{2}-6}\) | \(39\) |
Input:
int((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*ex p(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5 )+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3)*exp( -5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48* x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp( 5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(-4*x^5+ 48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x,method=_RETURNVERBOSE)
Output:
-1/(exp(-10*x/(exp(5)-2*x))+2*exp(-5*x/(exp(5)-2*x))*x+x^2-6)
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=-\frac {1}{x^{2} + 2 \, x e^{\left (\frac {5 \, x}{2 \, x - e^{5}}\right )} + e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )} - 6} \] Input:
integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x ^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x *exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3 )*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x ^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2 )*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(- 4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="fricas")
Output:
-1/(x^2 + 2*x*e^(5*x/(2*x - e^5)) + e^(10*x/(2*x - e^5)) - 6)
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=- \frac {1}{x^{2} + 2 x e^{- \frac {5 x}{- 2 x + e^{5}}} - 6 + e^{- \frac {10 x}{- 2 x + e^{5}}}} \] Input:
integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))**2+(2*exp(5)**2-18*x*exp(5)+8 *x**2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5)**2-8*x**2*exp(5)+8*x**3)/((exp(5) **2-4*x*exp(5)+4*x**2)*exp(-5*x/(exp(5)-2*x))**4+(4*x*exp(5)**2-16*x**2*ex p(5)+16*x**3)*exp(-5*x/(exp(5)-2*x))**3+((6*x**2-12)*exp(5)**2+(-24*x**3+4 8*x)*exp(5)+24*x**4-48*x**2)*exp(-5*x/(exp(5)-2*x))**2+((4*x**3-24*x)*exp( 5)**2+(-16*x**4+96*x**2)*exp(5)+16*x**5-96*x**3)*exp(-5*x/(exp(5)-2*x))+(x **4-12*x**2+36)*exp(5)**2+(-4*x**5+48*x**3-144*x)*exp(5)+4*x**6-48*x**4+14 4*x**2),x)
Output:
-1/(x**2 + 2*x*exp(-5*x/(-2*x + exp(5))) - 6 + exp(-10*x/(-2*x + exp(5))))
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=-\frac {1}{x^{2} + 2 \, x e^{\left (\frac {5 \, e^{5}}{2 \, {\left (2 \, x - e^{5}\right )}} + \frac {5}{2}\right )} + e^{\left (\frac {5 \, e^{5}}{2 \, x - e^{5}} + 5\right )} - 6} \] Input:
integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x ^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x *exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3 )*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x ^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2 )*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(- 4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="maxima")
Output:
-1/(x^2 + 2*x*e^(5/2*e^5/(2*x - e^5) + 5/2) + e^(5*e^5/(2*x - e^5) + 5) - 6)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (26) = 52\).
Time = 9.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 7.82 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=\frac {2 \, {\left (\frac {4 \, x e^{5}}{2 \, x - e^{5}} - \frac {4 \, x^{2} e^{5}}{{\left (2 \, x - e^{5}\right )}^{2}} - e^{5}\right )} e^{\left (-5\right )}}{\frac {x^{2} e^{10}}{{\left (2 \, x - e^{5}\right )}^{2}} - \frac {4 \, x e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )}}{2 \, x - e^{5}} + \frac {4 \, x^{2} e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )}}{{\left (2 \, x - e^{5}\right )}^{2}} - \frac {2 \, x e^{\left (\frac {5 \, x}{2 \, x - e^{5}} + 5\right )}}{2 \, x - e^{5}} + \frac {4 \, x^{2} e^{\left (\frac {5 \, x}{2 \, x - e^{5}} + 5\right )}}{{\left (2 \, x - e^{5}\right )}^{2}} + \frac {24 \, x}{2 \, x - e^{5}} - \frac {24 \, x^{2}}{{\left (2 \, x - e^{5}\right )}^{2}} + e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )} - 6} \] Input:
integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x ^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x *exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3 )*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x ^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2 )*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(- 4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="giac")
Output:
2*(4*x*e^5/(2*x - e^5) - 4*x^2*e^5/(2*x - e^5)^2 - e^5)*e^(-5)/(x^2*e^10/( 2*x - e^5)^2 - 4*x*e^(10*x/(2*x - e^5))/(2*x - e^5) + 4*x^2*e^(10*x/(2*x - e^5))/(2*x - e^5)^2 - 2*x*e^(5*x/(2*x - e^5) + 5)/(2*x - e^5) + 4*x^2*e^( 5*x/(2*x - e^5) + 5)/(2*x - e^5)^2 + 24*x/(2*x - e^5) - 24*x^2/(2*x - e^5) ^2 + e^(10*x/(2*x - e^5)) - 6)
Time = 7.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=-\frac {1}{{\mathrm {e}}^{\frac {10\,x}{2\,x-{\mathrm {e}}^5}}+2\,x\,{\mathrm {e}}^{\frac {5\,x}{2\,x-{\mathrm {e}}^5}}+x^2-6} \] Input:
int((2*x*exp(10) - 10*exp(5)*exp((10*x)/(2*x - exp(5))) + exp((5*x)/(2*x - exp(5)))*(2*exp(10) - 18*x*exp(5) + 8*x^2) - 8*x^2*exp(5) + 8*x^3)/(exp(( 20*x)/(2*x - exp(5)))*(exp(10) - 4*x*exp(5) + 4*x^2) + exp(10)*(x^4 - 12*x ^2 + 36) + exp((10*x)/(2*x - exp(5)))*(exp(5)*(48*x - 24*x^3) + exp(10)*(6 *x^2 - 12) - 48*x^2 + 24*x^4) + exp((15*x)/(2*x - exp(5)))*(4*x*exp(10) - 16*x^2*exp(5) + 16*x^3) - exp(5)*(144*x - 48*x^3 + 4*x^5) - exp((5*x)/(2*x - exp(5)))*(exp(10)*(24*x - 4*x^3) - exp(5)*(96*x^2 - 16*x^4) + 96*x^3 - 16*x^5) + 144*x^2 - 48*x^4 + 4*x^6),x)
Output:
-1/(exp((10*x)/(2*x - exp(5))) + 2*x*exp((5*x)/(2*x - exp(5))) + x^2 - 6)
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} \left (2 e^{10}-18 e^5 x+8 x^2\right )}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} \left (e^{10}-4 e^5 x+4 x^2\right )+e^{-\frac {15 x}{e^5-2 x}} \left (4 e^{10} x-16 e^5 x^2+16 x^3\right )+e^{10} \left (36-12 x^2+x^4\right )+e^5 \left (-144 x+48 x^3-4 x^5\right )+e^{-\frac {10 x}{e^5-2 x}} \left (-48 x^2+24 x^4+e^{10} \left (-12+6 x^2\right )+e^5 \left (48 x-24 x^3\right )\right )+e^{-\frac {5 x}{e^5-2 x}} \left (-96 x^3+16 x^5+e^{10} \left (-24 x+4 x^3\right )+e^5 \left (96 x^2-16 x^4\right )\right )} \, dx=-\frac {e^{\frac {10 x}{e^{5}-2 x}}}{e^{\frac {10 x}{e^{5}-2 x}} x^{2}-6 e^{\frac {10 x}{e^{5}-2 x}}+2 e^{\frac {5 x}{e^{5}-2 x}} x +1} \] Input:
int((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*ex p(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5 )+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3)*exp( -5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48* x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp( 5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(-4*x^5+ 48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x)
Output:
( - e**((10*x)/(e**5 - 2*x)))/(e**((10*x)/(e**5 - 2*x))*x**2 - 6*e**((10*x )/(e**5 - 2*x)) + 2*e**((5*x)/(e**5 - 2*x))*x + 1)