Integrand size = 205, antiderivative size = 33 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=\log \left (\left (-e^x+\frac {4}{5-e^4-\frac {3 e^{e^{-2+x}}}{x}+x}\right )^2\right ) \] Output:
ln((4/(5-3*exp(exp(-2+x))/x+x-exp(4))-exp(x))^2)
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(33)=66\).
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.30 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=\frac {2 \left (-e^2 \log \left (3 e^{e^{-2+x}}-5 x+e^4 x-x^2\right )+e^2 \log \left (3 e^{e^{-2+x}+x}+4 x-5 e^x x+e^{4+x} x-e^x x^2\right )\right )}{e^2} \] Input:
Integrate[(18*E^(2*E^(-2 + x) + x) + 8*x^2 + E^E^(-2 + x)*(24 - 24*E^(-2 + x)*x + E^x*(-60*x + 12*E^4*x - 12*x^2)) + E^x*(50*x^2 + 2*E^8*x^2 + 20*x^ 3 + 2*x^4 + E^4*(-20*x^2 - 4*x^3)))/(9*E^(2*E^(-2 + x) + x) - 20*x^2 + 4*E ^4*x^2 - 4*x^3 + E^E^(-2 + x)*(12*x + E^x*(-30*x + 6*E^4*x - 6*x^2)) + E^x *(25*x^2 + E^8*x^2 + 10*x^3 + x^4 + E^4*(-10*x^2 - 2*x^3))),x]
Output:
(2*(-(E^2*Log[3*E^E^(-2 + x) - 5*x + E^4*x - x^2]) + E^2*Log[3*E^(E^(-2 + x) + x) + 4*x - 5*E^x*x + E^(4 + x)*x - E^x*x^2]))/E^2
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.
| \(\displaystyle \int \frac {8 x^2+e^{e^{x-2}} \left (e^x \left (-12 x^2+12 e^4 x-60 x\right )-24 e^{x-2} x+24\right )+e^x \left (2 x^4+20 x^3+2 e^8 x^2+50 x^2+e^4 \left (-4 x^3-20 x^2\right )\right )+18 e^{x+2 e^{x-2}}}{-4 x^3+4 e^4 x^2-20 x^2+e^{e^{x-2}} \left (e^x \left (-6 x^2+6 e^4 x-30 x\right )+12 x\right )+e^x \left (x^4+10 x^3+e^8 x^2+25 x^2+e^4 \left (-2 x^3-10 x^2\right )\right )+9 e^{x+2 e^{x-2}}} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^2+e^{e^{x-2}} \left (e^x \left (-12 x^2+12 e^4 x-60 x\right )-24 e^{x-2} x+24\right )+e^x \left (2 x^4+20 x^3+2 e^8 x^2+50 x^2+e^4 \left (-4 x^3-20 x^2\right )\right )+18 e^{x+2 e^{x-2}}}{-4 x^3+\left (4 e^4-20\right ) x^2+e^{e^{x-2}} \left (e^x \left (-6 x^2+6 e^4 x-30 x\right )+12 x\right )+e^x \left (x^4+10 x^3+e^8 x^2+25 x^2+e^4 \left (-2 x^3-10 x^2\right )\right )+9 e^{x+2 e^{x-2}}}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 x^2+e^{e^{x-2}} \left (e^x \left (-12 x^2+12 e^4 x-60 x\right )-24 e^{x-2} x+24\right )+e^x \left (2 x^4+20 x^3+2 e^8 x^2+50 x^2+e^4 \left (-4 x^3-20 x^2\right )\right )+18 e^{x+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right ) \left (-e^x x^2-5 \left (1-\frac {e^4}{5}\right ) e^x x+4 x+3 e^{x+e^{x-2}}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{e^{x-2}+2} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (e^4-10\right )\right ) x^2-12 \left (1-\frac {1}{2} e^2 \left (e^4-5\right )\right ) e^{e^{x-2}} x+9 e^{2 e^{x-2}+2}\right )}{e^2 \left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {8 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{e^{x-2}+2} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (e^4-11\right )\right ) x^3+12 \left (1-\frac {1}{2} e^2 \left (e^4-5\right )\right ) e^{e^{x-2}} x^2-9 e^{2 e^{x-2}+2} x-15 \left (1-\frac {e^4}{5}\right ) e^{e^{x-2}+2} x+9 e^{2 e^{x-2}+2}\right )}{e^2 \left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 \left (1-\frac {e^4}{5}\right ) e^x x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 \left (1-\frac {e^6}{2}\right ) e^{x+e^{x-2}} x+12 e^{e^{x-2}+2}+9 e^{x+2 e^{x-2}+2}\right )}{e^2 \left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-6 e^{x+e^{x-2}} \left (2-e^6\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x+10} x^2+e^{x+2} (x+5)^2 x^2-2 e^{x+6} (x+5) x^2+4 e^2 x^2-6 e^{x+e^{x-2}+2} (x+5) x-12 e^{x+e^{x-2}} \left (1-\frac {e^6}{2}\right ) x+12 e^{2+e^{x-2}}+9 e^{x+2 e^{x-2}+2}}{\left (-((x+5) x)+e^4 x+3 e^{e^{x-2}}\right ) \left (e^{x+4} x-e^x (x+5) x+4 x+3 e^{x+e^{x-2}}\right )}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {e^2 x^4+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3-6 e^{2+e^{x-2}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2-12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x+9 e^{2+2 e^{x-2}}}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2}+\frac {4 \left (-e^2 x^5-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4+6 e^{2+e^{x-2}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3+12 e^{e^{x-2}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2-9 e^{2+2 e^{x-2}} x-15 e^{2+e^{x-2}} \left (1-\frac {e^4}{5}\right ) x+9 e^{2+2 e^{x-2}}\right )}{\left (-x^2-5 \left (1-\frac {e^4}{5}\right ) x+3 e^{e^{x-2}}\right )^2 \left (-e^x x^2-5 e^x \left (1-\frac {e^4}{5}\right ) x+4 x+3 e^{x+e^{x-2}}\right )}\right )dx}{e^2}\) |
Input:
Int[(18*E^(2*E^(-2 + x) + x) + 8*x^2 + E^E^(-2 + x)*(24 - 24*E^(-2 + x)*x + E^x*(-60*x + 12*E^4*x - 12*x^2)) + E^x*(50*x^2 + 2*E^8*x^2 + 20*x^3 + 2* x^4 + E^4*(-20*x^2 - 4*x^3)))/(9*E^(2*E^(-2 + x) + x) - 20*x^2 + 4*E^4*x^2 - 4*x^3 + E^E^(-2 + x)*(12*x + E^x*(-30*x + 6*E^4*x - 6*x^2)) + E^x*(25*x ^2 + E^8*x^2 + 10*x^3 + x^4 + E^4*(-10*x^2 - 2*x^3))),x]
Output:
$Aborted
Time = 3.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(2 x +2 \ln \left ({\mathrm e}^{{\mathrm e}^{-2+x}}-\frac {x^{2}}{3}+\frac {x \,{\mathrm e}^{4}}{3}-\frac {5 x}{3}+\frac {4 x \,{\mathrm e}^{-x}}{3}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{-2+x}}-\frac {x^{2}}{3}-\frac {5 x}{3}+\frac {x \,{\mathrm e}^{4}}{3}\right )\) | \(56\) |
| parallelrisch | \(-2 \ln \left (-x \,{\mathrm e}^{4}+x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{-2+x}}+5 x \right )+2 \ln \left (-{\mathrm e}^{x} {\mathrm e}^{4} x +{\mathrm e}^{x} x^{2}-3 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{-2+x}}+5 \,{\mathrm e}^{x} x -4 x \right )\) | \(58\) |
| norman | \(-2 \ln \left (x \,{\mathrm e}^{4}-x^{2}-5 x +3 \,{\mathrm e}^{{\mathrm e}^{-2} {\mathrm e}^{x}}\right )+2 \ln \left ({\mathrm e}^{x} {\mathrm e}^{4} x -{\mathrm e}^{x} x^{2}-5 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{{\mathrm e}^{-2} {\mathrm e}^{x}} {\mathrm e}^{x}+4 x \right )\) | \(61\) |
Input:
int((18*exp(x)*exp(exp(-2+x))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp (-2+x)+24)*exp(exp(-2+x))+(2*x^2*exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20* x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(-2+x))^2+((6*x*exp(4)-6*x^2-30 *x)*exp(x)+12*x)*exp(exp(-2+x))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+1 0*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3-20*x^2),x,method=_RETURNVERBOSE)
Output:
2*x+2*ln(exp(exp(-2+x))-1/3*x^2+1/3*x*exp(4)-5/3*x+4/3*x*exp(-x))-2*ln(exp (exp(-2+x))-1/3*x^2-5/3*x+1/3*x*exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=2 \, x - 2 \, \log \left (-x^{2} + x e^{4} - 5 \, x + 3 \, e^{\left (e^{\left (x - 2\right )}\right )}\right ) + 2 \, \log \left (-{\left ({\left (x^{2} - x e^{4} + 5 \, x\right )} e^{x} - 4 \, x - 3 \, e^{\left (x + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-x\right )}\right ) \] Input:
integrate((18*exp(x)*exp(exp(-2+x))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24 *x*exp(-2+x)+24)*exp(exp(-2+x))+(2*x^2*exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x ^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(-2+x))^2+((6*x*exp(4)-6* x^2-30*x)*exp(x)+12*x)*exp(exp(-2+x))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4) +x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3-20*x^2),x, algorithm="fricas ")
Output:
2*x - 2*log(-x^2 + x*e^4 - 5*x + 3*e^(e^(x - 2))) + 2*log(-((x^2 - x*e^4 + 5*x)*e^x - 4*x - 3*e^(x + e^(x - 2)))*e^(-x))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).
Time = 0.90 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.52 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=2 x - 2 \log {\left (\frac {\left (- x^{2} e^{x} - 5 x e^{x} + x e^{4} e^{x}\right ) e^{- x}}{3} + e^{\frac {e^{x}}{e^{2}}} \right )} + 2 \log {\left (\frac {\left (- x^{2} e^{x} - 5 x e^{x} + x e^{4} e^{x} + 4 x\right ) e^{- x}}{3} + e^{\frac {e^{x}}{e^{2}}} \right )} \] Input:
integrate((18*exp(x)*exp(exp(-2+x))**2+((12*x*exp(4)-12*x**2-60*x)*exp(x)- 24*x*exp(-2+x)+24)*exp(exp(-2+x))+(2*x**2*exp(4)**2+(-4*x**3-20*x**2)*exp( 4)+2*x**4+20*x**3+50*x**2)*exp(x)+8*x**2)/(9*exp(x)*exp(exp(-2+x))**2+((6* x*exp(4)-6*x**2-30*x)*exp(x)+12*x)*exp(exp(-2+x))+(x**2*exp(4)**2+(-2*x**3 -10*x**2)*exp(4)+x**4+10*x**3+25*x**2)*exp(x)+4*x**2*exp(4)-4*x**3-20*x**2 ),x)
Output:
2*x - 2*log((-x**2*exp(x) - 5*x*exp(x) + x*exp(4)*exp(x))*exp(-x)/3 + exp( exp(-2)*exp(x))) + 2*log((-x**2*exp(x) - 5*x*exp(x) + x*exp(4)*exp(x) + 4* x)*exp(-x)/3 + exp(exp(-2)*exp(x)))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=2 \, x - 2 \, \log \left (-\frac {1}{3} \, x^{2} + \frac {1}{3} \, x {\left (e^{4} - 5\right )} + e^{\left (e^{\left (x - 2\right )}\right )}\right ) + 2 \, \log \left (-\frac {1}{3} \, {\left ({\left (x^{2} - x {\left (e^{4} - 5\right )}\right )} e^{x} - 4 \, x - 3 \, e^{\left (x + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-x\right )}\right ) \] Input:
integrate((18*exp(x)*exp(exp(-2+x))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24 *x*exp(-2+x)+24)*exp(exp(-2+x))+(2*x^2*exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x ^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(-2+x))^2+((6*x*exp(4)-6* x^2-30*x)*exp(x)+12*x)*exp(exp(-2+x))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4) +x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3-20*x^2),x, algorithm="maxima ")
Output:
2*x - 2*log(-1/3*x^2 + 1/3*x*(e^4 - 5) + e^(e^(x - 2))) + 2*log(-1/3*((x^2 - x*(e^4 - 5))*e^x - 4*x - 3*e^(x + e^(x - 2)))*e^(-x))
Timed out. \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=\text {Timed out} \] Input:
integrate((18*exp(x)*exp(exp(-2+x))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24 *x*exp(-2+x)+24)*exp(exp(-2+x))+(2*x^2*exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x ^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(-2+x))^2+((6*x*exp(4)-6* x^2-30*x)*exp(x)+12*x)*exp(exp(-2+x))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4) +x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3-20*x^2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=\int \frac {18\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^{x-2}}-{\mathrm {e}}^{{\mathrm {e}}^{x-2}}\,\left (24\,x\,{\mathrm {e}}^{x-2}+{\mathrm {e}}^x\,\left (60\,x-12\,x\,{\mathrm {e}}^4+12\,x^2\right )-24\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^8-{\mathrm {e}}^4\,\left (4\,x^3+20\,x^2\right )+50\,x^2+20\,x^3+2\,x^4\right )+8\,x^2}{9\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^{x-2}}+{\mathrm {e}}^x\,\left (x^2\,{\mathrm {e}}^8-{\mathrm {e}}^4\,\left (2\,x^3+10\,x^2\right )+25\,x^2+10\,x^3+x^4\right )+{\mathrm {e}}^{{\mathrm {e}}^{x-2}}\,\left (12\,x-{\mathrm {e}}^x\,\left (30\,x-6\,x\,{\mathrm {e}}^4+6\,x^2\right )\right )+4\,x^2\,{\mathrm {e}}^4-20\,x^2-4\,x^3} \,d x \] Input:
int((18*exp(2*exp(x - 2))*exp(x) - exp(exp(x - 2))*(24*x*exp(x - 2) + exp( x)*(60*x - 12*x*exp(4) + 12*x^2) - 24) + exp(x)*(2*x^2*exp(8) - exp(4)*(20 *x^2 + 4*x^3) + 50*x^2 + 20*x^3 + 2*x^4) + 8*x^2)/(exp(x)*(x^2*exp(8) - ex p(4)*(10*x^2 + 2*x^3) + 25*x^2 + 10*x^3 + x^4) + exp(exp(x - 2))*(12*x - e xp(x)*(30*x - 6*x*exp(4) + 6*x^2)) + 9*exp(2*exp(x - 2))*exp(x) + 4*x^2*ex p(4) - 20*x^2 - 4*x^3),x)
Output:
int((18*exp(x + 2*exp(x - 2)) - exp(exp(x - 2))*(24*x*exp(x - 2) + exp(x)* (60*x - 12*x*exp(4) + 12*x^2) - 24) + exp(x)*(2*x^2*exp(8) - exp(4)*(20*x^ 2 + 4*x^3) + 50*x^2 + 20*x^3 + 2*x^4) + 8*x^2)/(9*exp(x + 2*exp(x - 2)) + exp(x)*(x^2*exp(8) - exp(4)*(10*x^2 + 2*x^3) + 25*x^2 + 10*x^3 + x^4) + ex p(exp(x - 2))*(12*x - exp(x)*(30*x - 6*x*exp(4) + 6*x^2)) + 4*x^2*exp(4) - 20*x^2 - 4*x^3), x)
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx=-2 \,\mathrm {log}\left (3 e^{\frac {e^{x}}{e^{2}}}+e^{4} x -x^{2}-5 x \right )+2 \,\mathrm {log}\left (3 e^{\frac {e^{x}+e^{2} x}{e^{2}}}+e^{x} e^{4} x -e^{x} x^{2}-5 e^{x} x +4 x \right ) \] Input:
int((18*exp(x)*exp(exp(-2+x))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp (-2+x)+24)*exp(exp(-2+x))+(2*x^2*exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20* x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(-2+x))^2+((6*x*exp(4)-6*x^2-30 *x)*exp(x)+12*x)*exp(exp(-2+x))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+1 0*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3-20*x^2),x)
Output:
2*( - log(3*e**(e**x/e**2) + e**4*x - x**2 - 5*x) + log(3*e**((e**x + e**2 *x)/e**2) + e**x*e**4*x - e**x*x**2 - 5*e**x*x + 4*x))