Integrand size = 150, antiderivative size = 28 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=e^5+e^{\left (-1+\left (-2+e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )\right )^2} \] Output:
exp((-1+((5-x)*exp(x)-2)*ln(3/2*x))^2)+exp(5)
\[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=\int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx \] Input:
Integrate[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x ) + E^(2*x)*(25 - 10*x + x^2))*Log[(3*x)/2]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*(-40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-16 *x + 4*x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x,x]
Output:
Integrate[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x ) + E^(2*x)*(25 - 10*x + x^2))*Log[(3*x)/2]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*(-40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-16 *x + 4*x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x, 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.
| \(\displaystyle \int \frac {\left (\left (e^x \left (2 x^2-40\right )+e^{2 x} \left (2 x^2-20 x+50\right )+8\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (4 x^2-16 x\right )+e^{2 x} \left (2 x^3-18 x^2+40 x\right )\right ) \log ^2\left (\frac {3 x}{2}\right )+e^x (2 x-10)+4\right ) \exp \left (\left (e^{2 x} \left (x^2-10 x+25\right )+e^x (4 x-20)+4\right ) \log ^2\left (\frac {3 x}{2}\right )+\left (e^x (2 x-10)+4\right ) \log \left (\frac {3 x}{2}\right )+1\right )}{x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\left (e^x \left (2 x^2-40\right )+e^{2 x} \left (2 x^2-20 x+50\right )+8\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (4 x^2-16 x\right )+e^{2 x} \left (2 x^3-18 x^2+40 x\right )\right ) \log ^2\left (\frac {3 x}{2}\right )+e^x (2 x-10)+4\right ) \exp \left (\left (-5 e^x \log \left (\frac {3 x}{2}\right )+e^x x \log \left (\frac {3 x}{2}\right )+2 \log \left (\frac {3 x}{2}\right )+1\right )^2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2 \log ^2\left (\frac {3 x}{2}\right )+x^2 \log \left (\frac {3 x}{2}\right )+x-8 x \log ^2\left (\frac {3 x}{2}\right )-20 \log \left (\frac {3 x}{2}\right )-5\right ) \exp \left (x+\left (-5 e^x \log \left (\frac {3 x}{2}\right )+e^x x \log \left (\frac {3 x}{2}\right )+2 \log \left (\frac {3 x}{2}\right )+1\right )^2\right )}{x}+\frac {2 (x-5) \log \left (\frac {3 x}{2}\right ) \left (x^2 \log \left (\frac {3 x}{2}\right )+x-4 x \log \left (\frac {3 x}{2}\right )-5\right ) \exp \left (2 x+\left (-5 e^x \log \left (\frac {3 x}{2}\right )+e^x x \log \left (\frac {3 x}{2}\right )+2 \log \left (\frac {3 x}{2}\right )+1\right )^2\right )}{x}+\frac {4 \left (2 \log \left (\frac {3 x}{2}\right )+1\right ) \exp \left (\left (-5 e^x \log \left (\frac {3 x}{2}\right )+e^x x \log \left (\frac {3 x}{2}\right )+2 \log \left (\frac {3 x}{2}\right )+1\right )^2\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (\left (e^x \left (x^2-20\right )+e^{2 x} (x-5)^2+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+2\right )}{x}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (2-e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (-e^x \left (2-e^x (5-x)\right ) (4-x) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (5-x)^2-e^x \left (20-x^2\right )+4\right ) \log \left (\frac {3 x}{2}\right )-e^x (5-x)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (2-e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (2-e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (2-e^x (5-x)\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (2 \log \left (\frac {3 x}{2}\right )+1\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+2 x\right ) (x-5) \log \left (\frac {3 x}{2}\right ) \left (\log \left (\frac {3 x}{2}\right ) x^2-4 \log \left (\frac {3 x}{2}\right ) x+x-5\right )}{x}+\frac {\exp \left (\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2+x\right ) \left (2 \log ^2\left (\frac {3 x}{2}\right ) x^2+\log \left (\frac {3 x}{2}\right ) x^2-8 \log ^2\left (\frac {3 x}{2}\right ) x+x-20 \log \left (\frac {3 x}{2}\right )-5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{\left (\left (e^x (x-5)+2\right ) \log \left (\frac {3 x}{2}\right )+1\right )^2} \left (e^x \left (e^x (x-5)+2\right ) (x-4) x \log ^2\left (\frac {3 x}{2}\right )+\left (e^{2 x} (x-5)^2+e^x \left (x^2-20\right )+4\right ) \log \left (\frac {3 x}{2}\right )+e^x (x-5)+2\right )}{x}dx\) |
Input:
Int[(E^(1 + (4 + E^x*(-10 + 2*x))*Log[(3*x)/2] + (4 + E^x*(-20 + 4*x) + E^ (2*x)*(25 - 10*x + x^2))*Log[(3*x)/2]^2)*(4 + E^x*(-10 + 2*x) + (8 + E^x*( -40 + 2*x^2) + E^(2*x)*(50 - 20*x + 2*x^2))*Log[(3*x)/2] + (E^x*(-16*x + 4 *x^2) + E^(2*x)*(40*x - 18*x^2 + 2*x^3))*Log[(3*x)/2]^2))/x,x]
Output:
$Aborted
Timed out.
\[\int \frac {\left (\left (\left (2 x^{3}-18 x^{2}+40 x \right ) {\mathrm e}^{2 x}+\left (4 x^{2}-16 x \right ) {\mathrm e}^{x}\right ) \ln \left (\frac {3 x}{2}\right )^{2}+\left (\left (2 x^{2}-20 x +50\right ) {\mathrm e}^{2 x}+\left (2 x^{2}-40\right ) {\mathrm e}^{x}+8\right ) \ln \left (\frac {3 x}{2}\right )+\left (2 x -10\right ) {\mathrm e}^{x}+4\right ) {\mathrm e}^{\left (\left (x^{2}-10 x +25\right ) {\mathrm e}^{2 x}+\left (4 x -20\right ) {\mathrm e}^{x}+4\right ) \ln \left (\frac {3 x}{2}\right )^{2}+\left (\left (2 x -10\right ) {\mathrm e}^{x}+4\right ) \ln \left (\frac {3 x}{2}\right )+1}}{x}d x\]
Input:
int((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*ln(3/2*x)^2+((2*x^ 2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*ln(3/2*x)+(2*x-10)*exp(x)+4)*exp( ((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*ln(3/2*x)^2+((2*x-10)*exp(x)+4) *ln(3/2*x)+1)/x,x)
Output:
int((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*ln(3/2*x)^2+((2*x^ 2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*ln(3/2*x)+(2*x-10)*exp(x)+4)*exp( ((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*ln(3/2*x)^2+((2*x-10)*exp(x)+4) *ln(3/2*x)+1)/x,x)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=e^{\left ({\left ({\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x - 5\right )} e^{x} + 4\right )} \log \left (\frac {3}{2} \, x\right )^{2} + 2 \, {\left ({\left (x - 5\right )} e^{x} + 2\right )} \log \left (\frac {3}{2} \, x\right ) + 1\right )} \] Input:
integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2 +((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x) +4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10)* exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="fricas")
Output:
e^(((x^2 - 10*x + 25)*e^(2*x) + 4*(x - 5)*e^x + 4)*log(3/2*x)^2 + 2*((x - 5)*e^x + 2)*log(3/2*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 1.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=e^{\left (\left (2 x - 10\right ) e^{x} + 4\right ) \log {\left (\frac {3 x}{2} \right )} + \left (\left (4 x - 20\right ) e^{x} + \left (x^{2} - 10 x + 25\right ) e^{2 x} + 4\right ) \log {\left (\frac {3 x}{2} \right )}^{2} + 1} \] Input:
integrate((((2*x**3-18*x**2+40*x)*exp(x)**2+(4*x**2-16*x)*exp(x))*ln(3/2*x )**2+((2*x**2-20*x+50)*exp(x)**2+(2*x**2-40)*exp(x)+8)*ln(3/2*x)+(2*x-10)* exp(x)+4)*exp(((x**2-10*x+25)*exp(x)**2+(4*x-20)*exp(x)+4)*ln(3/2*x)**2+(( 2*x-10)*exp(x)+4)*ln(3/2*x)+1)/x,x)
Output:
exp(((2*x - 10)*exp(x) + 4)*log(3*x/2) + ((4*x - 20)*exp(x) + (x**2 - 10*x + 25)*exp(2*x) + 4)*log(3*x/2)**2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (21) = 42\).
Time = 2.19 (sec) , antiderivative size = 388, normalized size of antiderivative = 13.86 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx =\text {Too large to display} \] Input:
integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2 +((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x) +4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10)* exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="maxima")
Output:
81*2^(-8*log(3) - 4)*x^4*e^(x^2*e^(2*x)*log(3)^2 - 2*x^2*e^(2*x)*log(3)*lo g(2) + x^2*e^(2*x)*log(2)^2 + 2*x^2*e^(2*x)*log(3)*log(x) - 2*x^2*e^(2*x)* log(2)*log(x) + x^2*e^(2*x)*log(x)^2 - 10*x*e^(2*x)*log(3)^2 + 4*x*e^x*log (3)^2 + 20*x*e^(2*x)*log(3)*log(2) - 8*x*e^x*log(3)*log(2) - 10*x*e^(2*x)* log(2)^2 + 4*x*e^x*log(2)^2 - 20*x*e^(2*x)*log(3)*log(x) + 8*x*e^x*log(3)* log(x) + 20*x*e^(2*x)*log(2)*log(x) - 8*x*e^x*log(2)*log(x) - 10*x*e^(2*x) *log(x)^2 + 4*x*e^x*log(x)^2 + 2*x*e^x*log(3) + 25*e^(2*x)*log(3)^2 - 20*e ^x*log(3)^2 - 2*x*e^x*log(2) - 50*e^(2*x)*log(3)*log(2) + 40*e^x*log(3)*lo g(2) + 25*e^(2*x)*log(2)^2 - 20*e^x*log(2)^2 + 2*x*e^x*log(x) + 50*e^(2*x) *log(3)*log(x) - 40*e^x*log(3)*log(x) - 50*e^(2*x)*log(2)*log(x) + 40*e^x* log(2)*log(x) + 25*e^(2*x)*log(x)^2 - 20*e^x*log(x)^2 - 10*e^x*log(3) + 4* log(3)^2 + 10*e^x*log(2) + 4*log(2)^2 - 10*e^x*log(x) + 8*log(3)*log(x) - 8*log(2)*log(x) + 4*log(x)^2 + 1)
\[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=\int { \frac {2 \, {\left ({\left ({\left (x^{3} - 9 \, x^{2} + 20 \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{x}\right )} \log \left (\frac {3}{2} \, x\right )^{2} + {\left (x - 5\right )} e^{x} + {\left ({\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{2} - 20\right )} e^{x} + 4\right )} \log \left (\frac {3}{2} \, x\right ) + 2\right )} e^{\left ({\left ({\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x - 5\right )} e^{x} + 4\right )} \log \left (\frac {3}{2} \, x\right )^{2} + 2 \, {\left ({\left (x - 5\right )} e^{x} + 2\right )} \log \left (\frac {3}{2} \, x\right ) + 1\right )}}{x} \,d x } \] Input:
integrate((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2 +((2*x^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x) +4)*exp(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10)* exp(x)+4)*log(3/2*x)+1)/x,x, algorithm="giac")
Output:
integrate(2*(((x^3 - 9*x^2 + 20*x)*e^(2*x) + 2*(x^2 - 4*x)*e^x)*log(3/2*x) ^2 + (x - 5)*e^x + ((x^2 - 10*x + 25)*e^(2*x) + (x^2 - 20)*e^x + 4)*log(3/ 2*x) + 2)*e^(((x^2 - 10*x + 25)*e^(2*x) + 4*(x - 5)*e^x + 4)*log(3/2*x)^2 + 2*((x - 5)*e^x + 2)*log(3/2*x) + 1)/x, x)
Time = 3.29 (sec) , antiderivative size = 429, normalized size of antiderivative = 15.32 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx =\text {Too large to display} \] Input:
int((exp(log((3*x)/2)*(exp(x)*(2*x - 10) + 4) + log((3*x)/2)^2*(exp(x)*(4* x - 20) + exp(2*x)*(x^2 - 10*x + 25) + 4) + 1)*(log((3*x)/2)^2*(exp(2*x)*( 40*x - 18*x^2 + 2*x^3) - exp(x)*(16*x - 4*x^2)) + exp(x)*(2*x - 10) + log( (3*x)/2)*(exp(2*x)*(2*x^2 - 20*x + 50) + exp(x)*(2*x^2 - 40) + 8) + 4))/x, x)
Output:
(81*2^(40*exp(x)*log(3))*2^(10*exp(x))*2^(20*x*exp(2*x)*log(3))*3^(2*x*exp (x))*x^(20*x*exp(2*x)*log(2))*x^(40*exp(x)*log(2))*x^(2*x*exp(x))*x^(8*log (3))*x^(2*x^2*exp(2*x)*log(3))*x^4*x^(50*exp(2*x)*log(3))*x^(8*x*exp(x)*lo g(3))*exp(4*log(x)^2)*exp(x^2*exp(2*x)*log(x)^2)*exp(-10*x*exp(2*x)*log(2) ^2)*exp(-10*x*exp(2*x)*log(3)^2)*exp(25*exp(2*x)*log(x)^2)*exp(4*x*exp(x)* log(x)^2)*exp(1)*exp(-20*exp(x)*log(2)^2)*exp(-20*exp(x)*log(3)^2)*exp(4*l og(2)^2)*exp(4*log(3)^2)*exp(x^2*exp(2*x)*log(2)^2)*exp(x^2*exp(2*x)*log(3 )^2)*exp(-10*x*exp(2*x)*log(x)^2)*exp(25*exp(2*x)*log(2)^2)*exp(25*exp(2*x )*log(3)^2)*exp(4*x*exp(x)*log(2)^2)*exp(4*x*exp(x)*log(3)^2)*exp(-20*exp( x)*log(x)^2))/(16*2^(2*x*exp(x))*2^(8*log(3))*2^(2*x^2*exp(2*x)*log(3))*2^ (50*exp(2*x)*log(3))*2^(8*x*exp(x)*log(3))*3^(10*exp(x))*x^(10*exp(x))*x^( 20*x*exp(2*x)*log(3))*x^(40*exp(x)*log(3))*x^(8*log(2))*x^(2*x^2*exp(2*x)* log(2))*x^(50*exp(2*x)*log(2))*x^(8*x*exp(x)*log(2)))
Time = 0.18 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {e^{1+\left (4+e^x (-10+2 x)\right ) \log \left (\frac {3 x}{2}\right )+\left (4+e^x (-20+4 x)+e^{2 x} \left (25-10 x+x^2\right )\right ) \log ^2\left (\frac {3 x}{2}\right )} \left (4+e^x (-10+2 x)+\left (8+e^x \left (-40+2 x^2\right )+e^{2 x} \left (50-20 x+2 x^2\right )\right ) \log \left (\frac {3 x}{2}\right )+\left (e^x \left (-16 x+4 x^2\right )+e^{2 x} \left (40 x-18 x^2+2 x^3\right )\right ) \log ^2\left (\frac {3 x}{2}\right )\right )}{x} \, dx=\frac {81 e^{e^{2 x} \mathrm {log}\left (\frac {3 x}{2}\right )^{2} x^{2}+25 e^{2 x} \mathrm {log}\left (\frac {3 x}{2}\right )^{2}+4 e^{x} \mathrm {log}\left (\frac {3 x}{2}\right )^{2} x +2 e^{x} \mathrm {log}\left (\frac {3 x}{2}\right ) x +4 \mathrm {log}\left (\frac {3 x}{2}\right )^{2}} e \,x^{4}}{16 e^{10 e^{2 x} \mathrm {log}\left (\frac {3 x}{2}\right )^{2} x +20 e^{x} \mathrm {log}\left (\frac {3 x}{2}\right )^{2}+10 e^{x} \mathrm {log}\left (\frac {3 x}{2}\right )}} \] Input:
int((((2*x^3-18*x^2+40*x)*exp(x)^2+(4*x^2-16*x)*exp(x))*log(3/2*x)^2+((2*x ^2-20*x+50)*exp(x)^2+(2*x^2-40)*exp(x)+8)*log(3/2*x)+(2*x-10)*exp(x)+4)*ex p(((x^2-10*x+25)*exp(x)^2+(4*x-20)*exp(x)+4)*log(3/2*x)^2+((2*x-10)*exp(x) +4)*log(3/2*x)+1)/x,x)
Output:
(81*e**(e**(2*x)*log((3*x)/2)**2*x**2 + 25*e**(2*x)*log((3*x)/2)**2 + 4*e* *x*log((3*x)/2)**2*x + 2*e**x*log((3*x)/2)*x + 4*log((3*x)/2)**2)*e*x**4)/ (16*e**(10*e**(2*x)*log((3*x)/2)**2*x + 20*e**x*log((3*x)/2)**2 + 10*e**x* log((3*x)/2)))