Integrand size = 122, antiderivative size = 34 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=-2+x+\left (5-e^x\right ) \left (\frac {e^{3-e^x}}{x (-5+9 x)}+\log (x)\right ) \] Output:
x+(ln(x)+exp(-exp(x)+3)/x/(9*x-5))*(5-exp(x))-2
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=x-\frac {e^{3-e^x} \left (-5+e^x\right )}{x (-5+9 x)}+5 \log (x)-e^x \log (x) \] Input:
Integrate[(125*x - 425*x^2 + 315*x^3 + 81*x^4 + E^x*(-25*x + 90*x^2 - 81*x ^3) + E^(3 - E^x)*(25 - 90*x + E^x*(-5 + 48*x - 54*x^2) + E^(2*x)*(-5*x + 9*x^2)) + E^x*(-25*x^2 + 90*x^3 - 81*x^4)*Log[x])/(25*x^2 - 90*x^3 + 81*x^ 4),x]
Output:
x - (E^(3 - E^x)*(-5 + E^x))/(x*(-5 + 9*x)) + 5*Log[x] - E^x*Log[x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {81 x^4+315 x^3-425 x^2+e^{3-e^x} \left (e^x \left (-54 x^2+48 x-5\right )+e^{2 x} \left (9 x^2-5 x\right )-90 x+25\right )+e^x \left (-81 x^3+90 x^2-25 x\right )+e^x \left (-81 x^4+90 x^3-25 x^2\right ) \log (x)+125 x}{81 x^4-90 x^3+25 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {81 x^4+315 x^3-425 x^2+e^{3-e^x} \left (e^x \left (-54 x^2+48 x-5\right )+e^{2 x} \left (9 x^2-5 x\right )-90 x+25\right )+e^x \left (-81 x^3+90 x^2-25 x\right )+e^x \left (-81 x^4+90 x^3-25 x^2\right ) \log (x)+125 x}{x^2 \left (81 x^2-90 x+25\right )}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 324 \int \frac {81 x^4+315 x^3-425 x^2+125 x-e^x \left (81 x^3-90 x^2+25 x\right )+e^{3-e^x} \left (-90 x-e^{2 x} \left (5 x-9 x^2\right )-e^x \left (54 x^2-48 x+5\right )+25\right )-e^x \left (81 x^4-90 x^3+25 x^2\right ) \log (x)}{324 (5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {81 x^4+315 x^3-425 x^2+e^{3-e^x} \left (-e^{2 x} \left (5 x-9 x^2\right )-e^x \left (54 x^2-48 x+5\right )-90 x+25\right )-e^x \left (81 x^3-90 x^2+25 x\right )-e^x \left (81 x^4-90 x^3+25 x^2\right ) \log (x)+125 x}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {81 x^2}{(9 x-5)^2}+\frac {25 e^{3-e^x}}{(9 x-5)^2 x^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{(9 x-5)^2 x^2}+\frac {315 x}{(9 x-5)^2}-\frac {425}{(9 x-5)^2}+\frac {e^{2 x-e^x+3}}{(9 x-5) x}-\frac {90 e^{3-e^x}}{(9 x-5)^2 x}+\frac {125}{(9 x-5)^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-e^x} \left (e^{x+3} \left (-54 x^2+48 x-5\right )-e^{x+e^x} x^2 (5-9 x)^2 \log (x)-e^{x+e^x} x (5-9 x)^2+e^{e^x} x (x+5) (5-9 x)^2+e^3 (25-90 x)+e^{2 x+3} x (9 x-5)\right )}{(5-9 x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{-e^x} \left (81 e^{e^x} x^4+315 e^{e^x} x^3-425 e^{e^x} x^2+125 e^{e^x} x-90 e^3 x+25 e^3\right )}{x^2 (9 x-5)^2}-\frac {e^{x-e^x} \left (81 e^{e^x} x^4 \log (x)+81 e^{e^x} x^3-90 e^{e^x} x^3 \log (x)-90 e^{e^x} x^2+54 e^3 x^2+25 e^{e^x} x^2 \log (x)+25 e^{e^x} x-48 e^3 x+5 e^3\right )}{x^2 (9 x-5)^2}+\frac {e^{2 x-e^x+3}}{x (9 x-5)}\right )dx\) |
Input:
Int[(125*x - 425*x^2 + 315*x^3 + 81*x^4 + E^x*(-25*x + 90*x^2 - 81*x^3) + E^(3 - E^x)*(25 - 90*x + E^x*(-5 + 48*x - 54*x^2) + E^(2*x)*(-5*x + 9*x^2) ) + E^x*(-25*x^2 + 90*x^3 - 81*x^4)*Log[x])/(25*x^2 - 90*x^3 + 81*x^4),x]
Output:
$Aborted
Time = 0.67 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(-{\mathrm e}^{x} \ln \left (x \right )+5 \ln \left (x \right )+x -\frac {\left ({\mathrm e}^{x}-5\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}}{x \left (9 x -5\right )}\) | \(36\) |
| parallelrisch | \(\frac {-405 x^{2} {\mathrm e}^{x} \ln \left (x \right )+2025 x^{2} \ln \left (x \right )+405 x^{3}+225 x \,{\mathrm e}^{x} \ln \left (x \right )-1125 x \ln \left (x \right )-45 \,{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{x}+3}-125 x +225 \,{\mathrm e}^{-{\mathrm e}^{x}+3}}{45 x \left (9 x -5\right )}\) | \(70\) |
Input:
int((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3) +(-81*x^4+90*x^3-25*x^2)*exp(x)*ln(x)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4+ 315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x,method=_RETURNVERBOSE)
Output:
-exp(x)*ln(x)+5*ln(x)+x-(exp(x)-5)/x/(9*x-5)*exp(-exp(x)+3)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.79 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=\frac {9 \, x^{3} - 5 \, x^{2} - {\left (e^{x} - 5\right )} e^{\left (-e^{x} + 3\right )} + {\left (45 \, x^{2} - {\left (9 \, x^{2} - 5 \, x\right )} e^{x} - 25 \, x\right )} \log \left (x\right )}{9 \, x^{2} - 5 \, x} \] Input:
integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp (x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+ 81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="fricas" )
Output:
(9*x^3 - 5*x^2 - (e^x - 5)*e^(-e^x + 3) + (45*x^2 - (9*x^2 - 5*x)*e^x - 25 *x)*log(x))/(9*x^2 - 5*x)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=x + \frac {\left (5 - e^{x}\right ) e^{3 - e^{x}}}{9 x^{2} - 5 x} - e^{x} \log {\left (x \right )} + 5 \log {\left (x \right )} \] Input:
integrate((((9*x**2-5*x)*exp(x)**2+(-54*x**2+48*x-5)*exp(x)-90*x+25)*exp(- exp(x)+3)+(-81*x**4+90*x**3-25*x**2)*exp(x)*ln(x)+(-81*x**3+90*x**2-25*x)* exp(x)+81*x**4+315*x**3-425*x**2+125*x)/(81*x**4-90*x**3+25*x**2),x)
Output:
x + (5 - exp(x))*exp(3 - exp(x))/(9*x**2 - 5*x) - exp(x)*log(x) + 5*log(x)
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=x - \frac {{\left (9 \, x^{2} - 5 \, x\right )} e^{x} \log \left (x\right ) - {\left (5 \, e^{3} - e^{\left (x + 3\right )}\right )} e^{\left (-e^{x}\right )}}{9 \, x^{2} - 5 \, x} + 5 \, \log \left (x\right ) \] Input:
integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp (x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+ 81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="maxima" )
Output:
x - ((9*x^2 - 5*x)*e^x*log(x) - (5*e^3 - e^(x + 3))*e^(-e^x))/(9*x^2 - 5*x ) + 5*log(x)
\[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=\int { \frac {81 \, x^{4} + 315 \, x^{3} - {\left (81 \, x^{4} - 90 \, x^{3} + 25 \, x^{2}\right )} e^{x} \log \left (x\right ) - 425 \, x^{2} - {\left (81 \, x^{3} - 90 \, x^{2} + 25 \, x\right )} e^{x} + {\left ({\left (9 \, x^{2} - 5 \, x\right )} e^{\left (2 \, x\right )} - {\left (54 \, x^{2} - 48 \, x + 5\right )} e^{x} - 90 \, x + 25\right )} e^{\left (-e^{x} + 3\right )} + 125 \, x}{81 \, x^{4} - 90 \, x^{3} + 25 \, x^{2}} \,d x } \] Input:
integrate((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp (x)+3)+(-81*x^4+90*x^3-25*x^2)*exp(x)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+ 81*x^4+315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x, algorithm="giac")
Output:
integrate((81*x^4 + 315*x^3 - (81*x^4 - 90*x^3 + 25*x^2)*e^x*log(x) - 425* x^2 - (81*x^3 - 90*x^2 + 25*x)*e^x + ((9*x^2 - 5*x)*e^(2*x) - (54*x^2 - 48 *x + 5)*e^x - 90*x + 25)*e^(-e^x + 3) + 125*x)/(81*x^4 - 90*x^3 + 25*x^2), x)
Time = 2.42 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=x+5\,\ln \left (x-\frac {5}{9}\right )+10\,\mathrm {atanh}\left (\frac {18\,x}{5}-1\right )-{\mathrm {e}}^x\,\ln \left (x\right )-\frac {{\mathrm {e}}^{3-{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x-5\right )}{x\,\left (9\,x-5\right )} \] Input:
int(-(exp(3 - exp(x))*(90*x + exp(2*x)*(5*x - 9*x^2) + exp(x)*(54*x^2 - 48 *x + 5) - 25) - 125*x + 425*x^2 - 315*x^3 - 81*x^4 + exp(x)*(25*x - 90*x^2 + 81*x^3) + exp(x)*log(x)*(25*x^2 - 90*x^3 + 81*x^4))/(25*x^2 - 90*x^3 + 81*x^4),x)
Output:
x + 5*log(x - 5/9) + 10*atanh((18*x)/5 - 1) - exp(x)*log(x) - (exp(3 - exp (x))*(exp(x) - 5))/(x*(9*x - 5))
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.94 \[ \int \frac {125 x-425 x^2+315 x^3+81 x^4+e^x \left (-25 x+90 x^2-81 x^3\right )+e^{3-e^x} \left (25-90 x+e^x \left (-5+48 x-54 x^2\right )+e^{2 x} \left (-5 x+9 x^2\right )\right )+e^x \left (-25 x^2+90 x^3-81 x^4\right ) \log (x)}{25 x^2-90 x^3+81 x^4} \, dx=\frac {-9 e^{e^{x}+x} \mathrm {log}\left (x \right ) x^{2}+5 e^{e^{x}+x} \mathrm {log}\left (x \right ) x +45 e^{e^{x}} \mathrm {log}\left (x \right ) x^{2}-25 e^{e^{x}} \mathrm {log}\left (x \right ) x +9 e^{e^{x}} x^{3}-5 e^{e^{x}} x^{2}-e^{x} e^{3}+5 e^{3}}{e^{e^{x}} x \left (9 x -5\right )} \] Input:
int((((9*x^2-5*x)*exp(x)^2+(-54*x^2+48*x-5)*exp(x)-90*x+25)*exp(-exp(x)+3) +(-81*x^4+90*x^3-25*x^2)*exp(x)*log(x)+(-81*x^3+90*x^2-25*x)*exp(x)+81*x^4 +315*x^3-425*x^2+125*x)/(81*x^4-90*x^3+25*x^2),x)
Output:
( - 9*e**(e**x + x)*log(x)*x**2 + 5*e**(e**x + x)*log(x)*x + 45*e**(e**x)* log(x)*x**2 - 25*e**(e**x)*log(x)*x + 9*e**(e**x)*x**3 - 5*e**(e**x)*x**2 - e**x*e**3 + 5*e**3)/(e**(e**x)*x*(9*x - 5))