Integrand size = 130, antiderivative size = 35 \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=\log \left (\frac {5 \log (5)}{-\frac {1}{5}+5 \left (x^2+\frac {(3-x)^2}{\left (x+\frac {x}{\log (x)}\right )^2}\right )}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(35)=70\).
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.26 \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=-50 \left (-\frac {\log (x)}{25}-\frac {1}{25} \log (1+\log (x))+\frac {1}{50} \log \left (-x^2+25 x^4-2 x^2 \log (x)+50 x^4 \log (x)+225 \log ^2(x)-150 x \log ^2(x)+24 x^2 \log ^2(x)+25 x^4 \log ^2(x)\right )\right ) \]
Integrate[(-50*x^4 + (-450 + 300*x - 50*x^2 - 150*x^4)*Log[x] + (450 - 150 *x - 150*x^4)*Log[x]^2 + (450 - 150*x - 50*x^4)*Log[x]^3)/(-x^3 + 25*x^5 + (-3*x^3 + 75*x^5)*Log[x] + (225*x - 150*x^2 + 22*x^3 + 75*x^5)*Log[x]^2 + (225*x - 150*x^2 + 24*x^3 + 25*x^5)*Log[x]^3),x]
-50*(-1/25*Log[x] - Log[1 + Log[x]]/25 + Log[-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4 *Log[x]^2]/50)
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.
| \(\displaystyle \int \frac {-50 x^4+\left (-50 x^4-150 x+450\right ) \log ^3(x)+\left (-150 x^4-150 x+450\right ) \log ^2(x)+\left (-150 x^4-50 x^2+300 x-450\right ) \log (x)}{25 x^5-x^3+\left (75 x^5-3 x^3\right ) \log (x)+\left (25 x^5+24 x^3-150 x^2+225 x\right ) \log ^3(x)+\left (75 x^5+22 x^3-150 x^2+225 x\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {50 x^4-\left (-50 x^4-150 x+450\right ) \log ^3(x)-\left (-150 x^4-150 x+450\right ) \log ^2(x)-\left (-150 x^4-50 x^2+300 x-450\right ) \log (x)}{x (\log (x)+1) \left (-25 x^4-25 x^4 \log ^2(x)-50 x^4 \log (x)+x^2-24 x^2 \log ^2(x)+2 x^2 \log (x)+150 x \log ^2(x)-225 \log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {50 \left (x^4+3 x-9\right )}{x \left (25 x^4+24 x^2-150 x+225\right )}-\frac {2 \left (625 x^8+625 x^8 \log (x)+1200 x^6+2450 x^6 \log (x)-9375 x^5-18750 x^5 \log (x)+16851 x^4+34326 x^4 \log (x)+225 x^3-7050 x^3 \log (x)-450 x^2+32850 x^2 \log (x)-67500 x \log (x)+50625 \log (x)\right )}{x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 x^4+25 x^4 \log ^2(x)+50 x^4 \log (x)-x^2+24 x^2 \log ^2(x)-2 x^2 \log (x)-150 x \log ^2(x)+225 \log ^2(x)\right )}+\frac {2}{x (\log (x)+1)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {50 \left (-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)\right )}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 50 \int \frac {x^4-\left (-x^4-3 x+9\right ) \log ^3(x)-3 \left (-x^4-x+3\right ) \log ^2(x)+\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (1-25 x^2\right ) x^2-\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+2 \left (x^2-25 x^4\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 50 \int \left (\frac {-x^4-3 x+9}{x \left (25 x^4+24 x^2-150 x+225\right )}+\frac {1}{25 x (\log (x)+1)}+\frac {-625 \log (x) x^8-625 x^8-2450 \log (x) x^6-1200 x^6+18750 \log (x) x^5+9375 x^5-34326 \log (x) x^4-16851 x^4+7050 \log (x) x^3-225 x^3-32850 \log (x) x^2+450 x^2+67500 \log (x) x-50625 \log (x)}{25 x \left (25 x^4+24 x^2-150 x+225\right ) \left (25 \log ^2(x) x^4+50 \log (x) x^4+25 x^4+24 \log ^2(x) x^2-2 \log (x) x^2-x^2-150 \log ^2(x) x+225 \log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 50 \int \frac {-x^4-\left (x^4+3 x-9\right ) \log ^3(x)-3 \left (x^4+x-3\right ) \log ^2(x)-\left (3 x^4+x^2-6 x+9\right ) \log (x)}{x (\log (x)+1) \left (\left (25 x^2-1\right ) x^2+\left (25 x^4+24 x^2-150 x+225\right ) \log ^2(x)+\left (50 x^4-2 x^2\right ) \log (x)\right )}dx\) |
Int[(-50*x^4 + (-450 + 300*x - 50*x^2 - 150*x^4)*Log[x] + (450 - 150*x - 1 50*x^4)*Log[x]^2 + (450 - 150*x - 50*x^4)*Log[x]^3)/(-x^3 + 25*x^5 + (-3*x ^3 + 75*x^5)*Log[x] + (225*x - 150*x^2 + 22*x^3 + 75*x^5)*Log[x]^2 + (225* x - 150*x^2 + 24*x^3 + 25*x^5)*Log[x]^3),x]
3.30.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(33)=66\).
Time = 0.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97
| method | result | size |
| parallelrisch | \(2 \ln \left (\ln \left (x \right )+1\right )-\ln \left (x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (x \right )+x^{4}+\frac {24 x^{2} \ln \left (x \right )^{2}}{25}-\frac {2 x^{2} \ln \left (x \right )}{25}-6 x \ln \left (x \right )^{2}-\frac {x^{2}}{25}+9 \ln \left (x \right )^{2}\right )+2 \ln \left (x \right )\) | \(69\) |
| default | \(2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )+1\right )-\ln \left (25 x^{4} \ln \left (x \right )^{2}+50 x^{4} \ln \left (x \right )+24 x^{2} \ln \left (x \right )^{2}+25 x^{4}-150 x \ln \left (x \right )^{2}-2 x^{2} \ln \left (x \right )+225 \ln \left (x \right )^{2}-x^{2}\right )\) | \(72\) |
| norman | \(2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )+1\right )-\ln \left (25 x^{4} \ln \left (x \right )^{2}+50 x^{4} \ln \left (x \right )+24 x^{2} \ln \left (x \right )^{2}+25 x^{4}-150 x \ln \left (x \right )^{2}-2 x^{2} \ln \left (x \right )+225 \ln \left (x \right )^{2}-x^{2}\right )\) | \(72\) |
| risch | \(2 \ln \left (x \right )-\ln \left (25 x^{4}+24 x^{2}-150 x +225\right )+2 \ln \left (\ln \left (x \right )+1\right )-\ln \left (\ln \left (x \right )^{2}+\frac {2 x^{2} \left (25 x^{2}-1\right ) \ln \left (x \right )}{25 x^{4}+24 x^{2}-150 x +225}+\frac {x^{2} \left (25 x^{2}-1\right )}{25 x^{4}+24 x^{2}-150 x +225}\right )\) | \(98\) |
int(((-50*x^4-150*x+450)*ln(x)^3+(-150*x^4-150*x+450)*ln(x)^2+(-150*x^4-50 *x^2+300*x-450)*ln(x)-50*x^4)/((25*x^5+24*x^3-150*x^2+225*x)*ln(x)^3+(75*x ^5+22*x^3-150*x^2+225*x)*ln(x)^2+(75*x^5-3*x^3)*ln(x)+25*x^5-x^3),x,method =_RETURNVERBOSE)
2*ln(ln(x)+1)-ln(x^4*ln(x)^2+2*x^4*ln(x)+x^4+24/25*x^2*ln(x)^2-2/25*x^2*ln (x)-6*x*ln(x)^2-1/25*x^2+9*ln(x)^2)+2*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (31) = 62\).
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.77 \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=-\log \left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right ) + 2 \, \log \left (x\right ) - \log \left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (25 \, x^{4} - x^{2}\right )} \log \left (x\right )}{25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225}\right ) + 2 \, \log \left (\log \left (x\right ) + 1\right ) \]
integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-15 0*x^4-50*x^2+300*x-450)*log(x)-50*x^4)/((25*x^5+24*x^3-150*x^2+225*x)*log( x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5-x ^3),x, algorithm=\
-log(25*x^4 + 24*x^2 - 150*x + 225) + 2*log(x) - log((25*x^4 + (25*x^4 + 2 4*x^2 - 150*x + 225)*log(x)^2 - x^2 + 2*(25*x^4 - x^2)*log(x))/(25*x^4 + 2 4*x^2 - 150*x + 225)) + 2*log(log(x) + 1)
Exception generated. \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((-50*x**4-150*x+450)*ln(x)**3+(-150*x**4-150*x+450)*ln(x)**2+(- 150*x**4-50*x**2+300*x-450)*ln(x)-50*x**4)/((25*x**5+24*x**3-150*x**2+225* x)*ln(x)**3+(75*x**5+22*x**3-150*x**2+225*x)*ln(x)**2+(75*x**5-3*x**3)*ln( x)+25*x**5-x**3),x)
Exception raised: PolynomialError >> 1/(390625*x**18 + 1500000*x**16 - 937 5000*x**15 + 16222500*x**14 - 27000000*x**13 + 126257400*x**12 - 279045000 *x**11 + 391055526*x**10 - 831794400*x**9 + 1973451600*x**8 - 2835405000*x **7 + 327827250
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (31) = 62\).
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.77 \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=-\log \left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right ) + 2 \, \log \left (x\right ) - \log \left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (25 \, x^{4} - x^{2}\right )} \log \left (x\right )}{25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225}\right ) + 2 \, \log \left (\log \left (x\right ) + 1\right ) \]
integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-15 0*x^4-50*x^2+300*x-450)*log(x)-50*x^4)/((25*x^5+24*x^3-150*x^2+225*x)*log( x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5-x ^3),x, algorithm=\
-log(25*x^4 + 24*x^2 - 150*x + 225) + 2*log(x) - log((25*x^4 + (25*x^4 + 2 4*x^2 - 150*x + 225)*log(x)^2 - x^2 + 2*(25*x^4 - x^2)*log(x))/(25*x^4 + 2 4*x^2 - 150*x + 225)) + 2*log(log(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).
Time = 0.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.03 \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=-\log \left (25 \, x^{4} \log \left (x\right )^{2} + 50 \, x^{4} \log \left (x\right ) + 25 \, x^{4} + 24 \, x^{2} \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x\right ) - 150 \, x \log \left (x\right )^{2} - x^{2} + 225 \, \log \left (x\right )^{2}\right ) + 2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x\right ) + 1\right ) \]
integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-15 0*x^4-50*x^2+300*x-450)*log(x)-50*x^4)/((25*x^5+24*x^3-150*x^2+225*x)*log( x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5-x ^3),x, algorithm=\
-log(25*x^4*log(x)^2 + 50*x^4*log(x) + 25*x^4 + 24*x^2*log(x)^2 - 2*x^2*lo g(x) - 150*x*log(x)^2 - x^2 + 225*log(x)^2) + 2*log(x) + 2*log(log(x) + 1)
Timed out. \[ \int \frac {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx=\int -\frac {{\ln \left (x\right )}^3\,\left (50\,x^4+150\,x-450\right )+{\ln \left (x\right )}^2\,\left (150\,x^4+150\,x-450\right )+50\,x^4+\ln \left (x\right )\,\left (150\,x^4+50\,x^2-300\,x+450\right )}{{\ln \left (x\right )}^3\,\left (25\,x^5+24\,x^3-150\,x^2+225\,x\right )-\ln \left (x\right )\,\left (3\,x^3-75\,x^5\right )+{\ln \left (x\right )}^2\,\left (75\,x^5+22\,x^3-150\,x^2+225\,x\right )-x^3+25\,x^5} \,d x \]
int(-(log(x)^3*(150*x + 50*x^4 - 450) + log(x)^2*(150*x + 150*x^4 - 450) + 50*x^4 + log(x)*(50*x^2 - 300*x + 150*x^4 + 450))/(log(x)^3*(225*x - 150* x^2 + 24*x^3 + 25*x^5) - log(x)*(3*x^3 - 75*x^5) + log(x)^2*(225*x - 150*x ^2 + 22*x^3 + 75*x^5) - x^3 + 25*x^5),x)