Integrand size = 185, antiderivative size = 32 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=2 e^{-\left (-3+\frac {\left (9-\log \left (\frac {5}{x}\right )\right ) (x+\log (x))}{x}\right )^2} x+\log (2) \] Output:
2/exp(((9-ln(5/x))*(x+ln(x))/x-3)^2)*x+ln(2)
Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(32)=64\).
Time = 0.35 (sec) , antiderivative size = 160, normalized size of antiderivative = 5.00 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=2\ 5^{12+\frac {6 \left (5 x \log (x)+3 \left (\log \left (\frac {5}{x}\right )+\log (x)\right ) \left (9+\log \left (\frac {5}{x}\right )+\log (x)\right )\right )}{x^2}} e^{-\frac {18 \log ^3\left (\frac {5}{x}\right )+162 \log \left (\frac {5}{x}\right ) \log (x)+\log ^2\left (\frac {5}{x}\right ) \left (162+x^2+36 \log (x)+\log ^2(x)\right )+9 \left (4 x^2+9 \log ^2(x)\right )}{x^2}} \left (\frac {1}{x}\right )^{\frac {6 \left (5 x \log (x)+3 \left (\log \left (\frac {5}{x}\right )+\log (x)\right ) \left (9+\log \left (\frac {5}{x}\right )+\log (x)\right )\right )}{x^2}} x^{-\frac {108+11 x+2 \log ^2\left (\frac {5}{x}\right )}{x}} \] Input:
Integrate[(-216*x - 22*x^2 + (60*x + 4*x^2)*Log[5/x] - 4*x*Log[5/x]^2 + (- 324 + 156*x + (72 - 52*x)*Log[5/x] + (-4 + 4*x)*Log[5/x]^2)*Log[x] + (288 - 68*Log[5/x] + 4*Log[5/x]^2)*Log[x]^2)/(E^((36*x^2 - 12*x^2*Log[5/x] + x^ 2*Log[5/x]^2 + (108*x - 30*x*Log[5/x] + 2*x*Log[5/x]^2)*Log[x] + (81 - 18* Log[5/x] + Log[5/x]^2)*Log[x]^2)/x^2)*x^2),x]
Output:
(2*5^(12 + (6*(5*x*Log[x] + 3*(Log[5/x] + Log[x])*(9 + Log[5/x] + Log[x])) )/x^2)*(x^(-1))^((6*(5*x*Log[x] + 3*(Log[5/x] + Log[x])*(9 + Log[5/x] + Lo g[x])))/x^2))/(E^((18*Log[5/x]^3 + 162*Log[5/x]*Log[x] + Log[5/x]^2*(162 + x^2 + 36*Log[x] + Log[x]^2) + 9*(4*x^2 + 9*Log[x]^2))/x^2)*x^((108 + 11*x + 2*Log[5/x]^2)/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.
| \(\displaystyle \int \frac {\left (-22 x^2+\left (4 x^2+60 x\right ) \log \left (\frac {5}{x}\right )-216 x-4 x \log ^2\left (\frac {5}{x}\right )+\left (4 \log ^2\left (\frac {5}{x}\right )-68 \log \left (\frac {5}{x}\right )+288\right ) \log ^2(x)+\left (156 x+(4 x-4) \log ^2\left (\frac {5}{x}\right )+(72-52 x) \log \left (\frac {5}{x}\right )-324\right ) \log (x)\right ) \exp \left (-\frac {36 x^2+x^2 \log ^2\left (\frac {5}{x}\right )-12 x^2 \log \left (\frac {5}{x}\right )+\left (\log ^2\left (\frac {5}{x}\right )-18 \log \left (\frac {5}{x}\right )+81\right ) \log ^2(x)+\left (108 x+2 x \log ^2\left (\frac {5}{x}\right )-30 x \log \left (\frac {5}{x}\right )\right ) \log (x)}{x^2}\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-22 x^2+\left (4 x^2+60 x\right ) \log \left (\frac {5}{x}\right )-216 x-4 x \log ^2\left (\frac {5}{x}\right )+\left (4 \log ^2\left (\frac {5}{x}\right )-68 \log \left (\frac {5}{x}\right )+288\right ) \log ^2(x)+\left (156 x+(4 x-4) \log ^2\left (\frac {5}{x}\right )+(72-52 x) \log \left (\frac {5}{x}\right )-324\right ) \log (x)\right ) \exp \left (-\frac {\left (-6 x+x \log \left (\frac {5}{x}\right )+\log \left (\frac {5}{x}\right ) \log (x)-9 \log (x)\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log \left (\frac {5}{x}\right )-9\right ) \left (\log \left (\frac {5}{x}\right )-8\right ) \log ^2(x) \exp \left (-\frac {\left (-6 x+x \log \left (\frac {5}{x}\right )+\log \left (\frac {5}{x}\right ) \log (x)-9 \log (x)\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (-6 x+x \log \left (\frac {5}{x}\right )+\log \left (\frac {5}{x}\right ) \log (x)-9 \log (x)\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (-6 x+x \log \left (\frac {5}{x}\right )+\log \left (\frac {5}{x}\right ) \log (x)-9 \log (x)\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x (11 x+108)+4 (\log (x)-1) (x+\log (x)) \log ^2\left (\frac {5}{x}\right )+4 \left (x (x+15)-17 \log ^2(x)+(18-13 x) \log (x)\right ) \log \left (\frac {5}{x}\right )+288 \log ^2(x)+12 (13 x-27) \log (x)\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (\log ^2\left (\frac {5}{x}\right )-17 \log \left (\frac {5}{x}\right )+72\right ) \log ^2(x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {4 \left (39 x+x \log ^2\left (\frac {5}{x}\right )-\log ^2\left (\frac {5}{x}\right )-13 x \log \left (\frac {5}{x}\right )+18 \log \left (\frac {5}{x}\right )-81\right ) \log (x) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x^2}+\frac {2 \left (-11 x-2 \log ^2\left (\frac {5}{x}\right )+2 x \log \left (\frac {5}{x}\right )+30 \log \left (\frac {5}{x}\right )-108\right ) \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log ^2\left (\frac {5}{x}\right )}{x}dx-4 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log ^2\left (\frac {5}{x}\right ) \log (x)}{x^2}dx+4 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log ^2\left (\frac {5}{x}\right ) \log (x)}{x}dx+288 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log ^2(x)}{x^2}dx-68 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log \left (\frac {5}{x}\right ) \log ^2(x)}{x^2}dx+4 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log ^2\left (\frac {5}{x}\right ) \log ^2(x)}{x^2}dx-22 \int \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )dx-216 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right )}{x}dx+4 \int \exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log \left (\frac {5}{x}\right )dx+60 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log \left (\frac {5}{x}\right )}{x}dx-324 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log (x)}{x^2}dx+156 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log (x)}{x}dx+72 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log \left (\frac {5}{x}\right ) \log (x)}{x^2}dx-52 \int \frac {\exp \left (-\frac {\left (6 x+9 \log (x)-\log \left (\frac {5}{x}\right ) (x+\log (x))\right )^2}{x^2}\right ) \log \left (\frac {5}{x}\right ) \log (x)}{x}dx\) |
Input:
Int[(-216*x - 22*x^2 + (60*x + 4*x^2)*Log[5/x] - 4*x*Log[5/x]^2 + (-324 + 156*x + (72 - 52*x)*Log[5/x] + (-4 + 4*x)*Log[5/x]^2)*Log[x] + (288 - 68*L og[5/x] + 4*Log[5/x]^2)*Log[x]^2)/(E^((36*x^2 - 12*x^2*Log[5/x] + x^2*Log[ 5/x]^2 + (108*x - 30*x*Log[5/x] + 2*x*Log[5/x]^2)*Log[x] + (81 - 18*Log[5/ x] + Log[5/x]^2)*Log[x]^2)/x^2)*x^2),x]
Output:
$Aborted
Time = 5.89 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(2 x \,{\mathrm e}^{-\frac {\left (\ln \left (5\right ) \ln \left (x \right )+x \ln \left (5\right )-\ln \left (x \right )^{2}-x \ln \left (x \right )-9 \ln \left (x \right )-6 x \right )^{2}}{x^{2}}}\) | \(40\) |
| parallelrisch | \(2 x \,{\mathrm e}^{-\frac {\left (\ln \left (\frac {5}{x}\right )^{2}-18 \ln \left (\frac {5}{x}\right )+81\right ) \ln \left (x \right )^{2}+\left (2 x \ln \left (\frac {5}{x}\right )^{2}-30 x \ln \left (\frac {5}{x}\right )+108 x \right ) \ln \left (x \right )+x^{2} \ln \left (\frac {5}{x}\right )^{2}-12 x^{2} \ln \left (\frac {5}{x}\right )+36 x^{2}}{x^{2}}}\) | \(90\) |
Input:
int(((4*ln(5/x)^2-68*ln(5/x)+288)*ln(x)^2+((-4+4*x)*ln(5/x)^2+(-52*x+72)*l n(5/x)+156*x-324)*ln(x)-4*x*ln(5/x)^2+(4*x^2+60*x)*ln(5/x)-22*x^2-216*x)/x ^2/exp(((ln(5/x)^2-18*ln(5/x)+81)*ln(x)^2+(2*x*ln(5/x)^2-30*x*ln(5/x)+108* x)*ln(x)+x^2*ln(5/x)^2-12*x^2*ln(5/x)+36*x^2)/x^2),x,method=_RETURNVERBOSE )
Output:
2*x*exp(-(ln(5)*ln(x)+x*ln(5)-ln(x)^2-x*ln(x)-9*ln(x)-6*x)^2/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (29) = 58\).
Time = 10.60 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=2 \, x e^{\left (\frac {2 \, {\left (x + \log \left (5\right ) + 9\right )} \log \left (\frac {5}{x}\right )^{3} - \log \left (\frac {5}{x}\right )^{4} - {\left (x^{2} + 2 \, {\left (x + 18\right )} \log \left (5\right ) + \log \left (5\right )^{2} + 30 \, x + 81\right )} \log \left (\frac {5}{x}\right )^{2} - 36 \, x^{2} - 108 \, x \log \left (5\right ) - 81 \, \log \left (5\right )^{2} + 6 \, {\left (2 \, x^{2} + {\left (5 \, x + 27\right )} \log \left (5\right ) + 3 \, \log \left (5\right )^{2} + 18 \, x\right )} \log \left (\frac {5}{x}\right )}{x^{2}}\right )} \] Input:
integrate(((4*log(5/x)^2-68*log(5/x)+288)*log(x)^2+((-4+4*x)*log(5/x)^2+(- 52*x+72)*log(5/x)+156*x-324)*log(x)-4*x*log(5/x)^2+(4*x^2+60*x)*log(5/x)-2 2*x^2-216*x)/x^2/exp(((log(5/x)^2-18*log(5/x)+81)*log(x)^2+(2*x*log(5/x)^2 -30*x*log(5/x)+108*x)*log(x)+x^2*log(5/x)^2-12*x^2*log(5/x)+36*x^2)/x^2),x , algorithm="fricas")
Output:
2*x*e^((2*(x + log(5) + 9)*log(5/x)^3 - log(5/x)^4 - (x^2 + 2*(x + 18)*log (5) + log(5)^2 + 30*x + 81)*log(5/x)^2 - 36*x^2 - 108*x*log(5) - 81*log(5) ^2 + 6*(2*x^2 + (5*x + 27)*log(5) + 3*log(5)^2 + 18*x)*log(5/x))/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.67 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=2 x e^{- \frac {x^{2} \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} - 12 x^{2} \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right ) + 36 x^{2} + \left (2 x \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} - 30 x \left (- \log {\left (x \right )} + \log {\left (5 \right )}\right ) + 108 x\right ) \log {\left (x \right )} + \left (\left (- \log {\left (x \right )} + \log {\left (5 \right )}\right )^{2} + 18 \log {\left (x \right )} - 18 \log {\left (5 \right )} + 81\right ) \log {\left (x \right )}^{2}}{x^{2}}} \] Input:
integrate(((4*ln(5/x)**2-68*ln(5/x)+288)*ln(x)**2+((-4+4*x)*ln(5/x)**2+(-5 2*x+72)*ln(5/x)+156*x-324)*ln(x)-4*x*ln(5/x)**2+(4*x**2+60*x)*ln(5/x)-22*x **2-216*x)/x**2/exp(((ln(5/x)**2-18*ln(5/x)+81)*ln(x)**2+(2*x*ln(5/x)**2-3 0*x*ln(5/x)+108*x)*ln(x)+x**2*ln(5/x)**2-12*x**2*ln(5/x)+36*x**2)/x**2),x)
Output:
2*x*exp(-(x**2*(-log(x) + log(5))**2 - 12*x**2*(-log(x) + log(5)) + 36*x** 2 + (2*x*(-log(x) + log(5))**2 - 30*x*(-log(x) + log(5)) + 108*x)*log(x) + ((-log(x) + log(5))**2 + 18*log(x) - 18*log(5) + 81)*log(x)**2)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (29) = 58\).
Time = 0.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.50 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=\frac {488281250 \, e^{\left (-\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) - \frac {2 \, \log \left (5\right )^{2} \log \left (x\right )}{x} + \frac {4 \, \log \left (5\right ) \log \left (x\right )^{2}}{x} - \frac {\log \left (5\right )^{2} \log \left (x\right )^{2}}{x^{2}} - \log \left (x\right )^{2} - \frac {2 \, \log \left (x\right )^{3}}{x} + \frac {2 \, \log \left (5\right ) \log \left (x\right )^{3}}{x^{2}} - \frac {\log \left (x\right )^{4}}{x^{2}} + \frac {30 \, \log \left (5\right ) \log \left (x\right )}{x} - \frac {30 \, \log \left (x\right )^{2}}{x} + \frac {18 \, \log \left (5\right ) \log \left (x\right )^{2}}{x^{2}} - \frac {18 \, \log \left (x\right )^{3}}{x^{2}} - \frac {108 \, \log \left (x\right )}{x} - \frac {81 \, \log \left (x\right )^{2}}{x^{2}} - 36\right )}}{x^{11}} \] Input:
integrate(((4*log(5/x)^2-68*log(5/x)+288)*log(x)^2+((-4+4*x)*log(5/x)^2+(- 52*x+72)*log(5/x)+156*x-324)*log(x)-4*x*log(5/x)^2+(4*x^2+60*x)*log(5/x)-2 2*x^2-216*x)/x^2/exp(((log(5/x)^2-18*log(5/x)+81)*log(x)^2+(2*x*log(5/x)^2 -30*x*log(5/x)+108*x)*log(x)+x^2*log(5/x)^2-12*x^2*log(5/x)+36*x^2)/x^2),x , algorithm="maxima")
Output:
488281250*e^(-log(5)^2 + 2*log(5)*log(x) - 2*log(5)^2*log(x)/x + 4*log(5)* log(x)^2/x - log(5)^2*log(x)^2/x^2 - log(x)^2 - 2*log(x)^3/x + 2*log(5)*lo g(x)^3/x^2 - log(x)^4/x^2 + 30*log(5)*log(x)/x - 30*log(x)^2/x + 18*log(5) *log(x)^2/x^2 - 18*log(x)^3/x^2 - 108*log(x)/x - 81*log(x)^2/x^2 - 36)/x^1 1
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (29) = 58\).
Time = 0.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.34 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=2 \, x e^{\left (-\frac {x^{2} \log \left (5\right )^{2} - 2 \, x^{2} \log \left (5\right ) \log \left (x\right ) + 2 \, x \log \left (5\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 4 \, x \log \left (5\right ) \log \left (x\right )^{2} + \log \left (5\right )^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{3} - 2 \, \log \left (5\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} - 12 \, x^{2} \log \left (5\right ) + 12 \, x^{2} \log \left (x\right ) - 30 \, x \log \left (5\right ) \log \left (x\right ) + 30 \, x \log \left (x\right )^{2} - 18 \, \log \left (5\right ) \log \left (x\right )^{2} + 18 \, \log \left (x\right )^{3} + 36 \, x^{2} + 108 \, x \log \left (x\right ) + 81 \, \log \left (x\right )^{2}}{x^{2}}\right )} \] Input:
integrate(((4*log(5/x)^2-68*log(5/x)+288)*log(x)^2+((-4+4*x)*log(5/x)^2+(- 52*x+72)*log(5/x)+156*x-324)*log(x)-4*x*log(5/x)^2+(4*x^2+60*x)*log(5/x)-2 2*x^2-216*x)/x^2/exp(((log(5/x)^2-18*log(5/x)+81)*log(x)^2+(2*x*log(5/x)^2 -30*x*log(5/x)+108*x)*log(x)+x^2*log(5/x)^2-12*x^2*log(5/x)+36*x^2)/x^2),x , algorithm="giac")
Output:
2*x*e^(-(x^2*log(5)^2 - 2*x^2*log(5)*log(x) + 2*x*log(5)^2*log(x) + x^2*lo g(x)^2 - 4*x*log(5)*log(x)^2 + log(5)^2*log(x)^2 + 2*x*log(x)^3 - 2*log(5) *log(x)^3 + log(x)^4 - 12*x^2*log(5) + 12*x^2*log(x) - 30*x*log(5)*log(x) + 30*x*log(x)^2 - 18*log(5)*log(x)^2 + 18*log(x)^3 + 36*x^2 + 108*x*log(x) + 81*log(x)^2)/x^2)
Time = 4.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.81 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=\frac {488281250\,5^{\frac {18\,{\ln \left (x\right )}^2}{x^2}}\,x^{\frac {30\,\ln \left (\frac {1}{x}\right )}{x}}\,x^{\frac {30\,\ln \left (5\right )}{x}}\,{\mathrm {e}}^{-36}\,{\mathrm {e}}^{-{\ln \left (\frac {1}{x}\right )}^2}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x}\right )}^2\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-\frac {{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {81\,{\ln \left (x\right )}^2}{x^2}}\,{\left (\frac {1}{x}\right )}^{\frac {18\,{\ln \left (x\right )}^2}{x^2}}}{x^{108/x}\,x^{\frac {2\,{\ln \left (\frac {1}{x}\right )}^2}{x}}\,x^{\frac {4\,\ln \left (\frac {1}{x}\right )\,\ln \left (5\right )}{x}}\,x^{11}\,x^{\frac {2\,{\ln \left (5\right )}^2}{x}}\,{\left (\frac {1}{x}\right )}^{2\,\ln \left (5\right )}\,{\left (\frac {1}{x}\right )}^{\frac {2\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x^2}}} \] Input:
int(-(exp(-(log(x)^2*(log(5/x)^2 - 18*log(5/x) + 81) + x^2*log(5/x)^2 + lo g(x)*(108*x - 30*x*log(5/x) + 2*x*log(5/x)^2) + 36*x^2 - 12*x^2*log(5/x))/ x^2)*(216*x - log(x)*(156*x - log(5/x)*(52*x - 72) + log(5/x)^2*(4*x - 4) - 324) - log(x)^2*(4*log(5/x)^2 - 68*log(5/x) + 288) - log(5/x)*(60*x + 4* x^2) + 22*x^2 + 4*x*log(5/x)^2))/x^2,x)
Output:
(488281250*5^((18*log(x)^2)/x^2)*x^((30*log(1/x))/x)*x^((30*log(5))/x)*exp (-36)*exp(-log(1/x)^2)*exp(-(log(1/x)^2*log(x)^2)/x^2)*exp(-log(5)^2)*exp( -(log(5)^2*log(x)^2)/x^2)*exp(-(81*log(x)^2)/x^2)*(1/x)^((18*log(x)^2)/x^2 ))/(x^(108/x)*x^((2*log(1/x)^2)/x)*x^((4*log(1/x)*log(5))/x)*x^11*x^((2*lo g(5)^2)/x)*(1/x)^(2*log(5))*(1/x)^((2*log(5)*log(x)^2)/x^2))
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.00 \[ \int \frac {e^{-\frac {36 x^2-12 x^2 \log \left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )+\left (108 x-30 x \log \left (\frac {5}{x}\right )+2 x \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (81-18 \log \left (\frac {5}{x}\right )+\log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)}{x^2}} \left (-216 x-22 x^2+\left (60 x+4 x^2\right ) \log \left (\frac {5}{x}\right )-4 x \log ^2\left (\frac {5}{x}\right )+\left (-324+156 x+(72-52 x) \log \left (\frac {5}{x}\right )+(-4+4 x) \log ^2\left (\frac {5}{x}\right )\right ) \log (x)+\left (288-68 \log \left (\frac {5}{x}\right )+4 \log ^2\left (\frac {5}{x}\right )\right ) \log ^2(x)\right )}{x^2} \, dx=\frac {488281250 e^{\frac {18 \,\mathrm {log}\left (\frac {5}{x}\right ) \mathrm {log}\left (x \right )^{2}+30 \,\mathrm {log}\left (\frac {5}{x}\right ) \mathrm {log}\left (x \right ) x}{x^{2}}}}{e^{\frac {\mathrm {log}\left (\frac {5}{x}\right )^{2} \mathrm {log}\left (x \right )^{2}+2 \mathrm {log}\left (\frac {5}{x}\right )^{2} \mathrm {log}\left (x \right ) x +\mathrm {log}\left (\frac {5}{x}\right )^{2} x^{2}+81 \mathrm {log}\left (x \right )^{2}+108 \,\mathrm {log}\left (x \right ) x}{x^{2}}} e^{36} x^{11}} \] Input:
int(((4*log(5/x)^2-68*log(5/x)+288)*log(x)^2+((-4+4*x)*log(5/x)^2+(-52*x+7 2)*log(5/x)+156*x-324)*log(x)-4*x*log(5/x)^2+(4*x^2+60*x)*log(5/x)-22*x^2- 216*x)/x^2/exp(((log(5/x)^2-18*log(5/x)+81)*log(x)^2+(2*x*log(5/x)^2-30*x* log(5/x)+108*x)*log(x)+x^2*log(5/x)^2-12*x^2*log(5/x)+36*x^2)/x^2),x)
Output:
(488281250*e**((18*log(5/x)*log(x)**2 + 30*log(5/x)*log(x)*x)/x**2))/(e**( (log(5/x)**2*log(x)**2 + 2*log(5/x)**2*log(x)*x + log(5/x)**2*x**2 + 81*lo g(x)**2 + 108*log(x)*x)/x**2)*e**36*x**11)