Integrand size = 262, antiderivative size = 30 \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\left (5+\log \left (\frac {x}{3 \left (-\frac {1}{5}+x\right )^2}-\log \left (e^3 \left (4+x^2\right )\right )\right )\right )^4 \] Output:
(5+ln(1/3*x/(x-1/5)^2-ln((x^2+4)*exp(3))))^4
Leaf count is larger than twice the leaf count of optimal. \(169\) vs. \(2(30)=60\).
Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.63 \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=4 \left (125 \log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+\frac {75}{2} \log ^2\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+5 \log ^3\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+\frac {1}{4} \log ^4\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )\right ) \] Input:
Integrate[(50000 + 247000*x + 57500*x^2 - 162500*x^3 + 375000*x^4 + (30000 + 148200*x + 34500*x^2 - 97500*x^3 + 225000*x^4)*Log[(25*x + (-3 + 30*x - 75*x^2)*Log[E^3*(4 + x^2)])/(3 - 30*x + 75*x^2)] + (6000 + 29640*x + 6900 *x^2 - 19500*x^3 + 45000*x^4)*Log[(25*x + (-3 + 30*x - 75*x^2)*Log[E^3*(4 + x^2)])/(3 - 30*x + 75*x^2)]^2 + (400 + 1976*x + 460*x^2 - 1300*x^3 + 300 0*x^4)*Log[(25*x + (-3 + 30*x - 75*x^2)*Log[E^3*(4 + x^2)])/(3 - 30*x + 75 *x^2)]^3)/(100*x - 500*x^2 + 25*x^3 - 125*x^4 + (-12 + 180*x - 903*x^2 + 1 545*x^3 - 225*x^4 + 375*x^5)*Log[E^3*(4 + x^2)]),x]
Output:
4*(125*Log[-1/3*(9 - 115*x + 225*x^2 + 3*(1 - 5*x)^2*Log[4 + x^2])/(1 - 5* x)^2] + (75*Log[-1/3*(9 - 115*x + 225*x^2 + 3*(1 - 5*x)^2*Log[4 + x^2])/(1 - 5*x)^2]^2)/2 + 5*Log[-1/3*(9 - 115*x + 225*x^2 + 3*(1 - 5*x)^2*Log[4 + x^2])/(1 - 5*x)^2]^3 + Log[-1/3*(9 - 115*x + 225*x^2 + 3*(1 - 5*x)^2*Log[4 + x^2])/(1 - 5*x)^2]^4/4)
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.
| \(\displaystyle \int \frac {375000 x^4-162500 x^3+57500 x^2+\left (3000 x^4-1300 x^3+460 x^2+1976 x+400\right ) \log ^3\left (\frac {\left (-75 x^2+30 x-3\right ) \log \left (e^3 \left (x^2+4\right )\right )+25 x}{75 x^2-30 x+3}\right )+\left (45000 x^4-19500 x^3+6900 x^2+29640 x+6000\right ) \log ^2\left (\frac {\left (-75 x^2+30 x-3\right ) \log \left (e^3 \left (x^2+4\right )\right )+25 x}{75 x^2-30 x+3}\right )+\left (225000 x^4-97500 x^3+34500 x^2+148200 x+30000\right ) \log \left (\frac {\left (-75 x^2+30 x-3\right ) \log \left (e^3 \left (x^2+4\right )\right )+25 x}{75 x^2-30 x+3}\right )+247000 x+50000}{-125 x^4+25 x^3-500 x^2+\left (375 x^5-225 x^4+1545 x^3-903 x^2+180 x-12\right ) \log \left (e^3 \left (x^2+4\right )\right )+100 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (-750 x^4+325 x^3-115 x^2-494 x-100\right ) \left (\log \left (-\frac {225 x^2+3 (1-5 x)^2 \log \left (x^2+4\right )-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{\left (-5 x^3+x^2-20 x+4\right ) \left (225 x^2+3 (1-5 x)^2 \log \left (x^2+4\right )-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{\left (-5 x^3+x^2-20 x+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{\left (-5 x^3+x^2-20 x+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle -4 \int \left (\frac {(5 x+1) \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{101 \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}-\frac {25 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{101 (5 x-1) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \left (\log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )+5\right )^3}{(1-5 x) \left (x^2+4\right ) \left (3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -4 \int \left (-\frac {\left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^3\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {15 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log ^2\left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {75 \left (750 x^4-325 x^3+115 x^2+494 x+100\right ) \log \left (-\frac {3 \log \left (x^2+4\right ) (1-5 x)^2+225 x^2-115 x+9}{3 (1-5 x)^2}\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}-\frac {125 \left (750 x^4-325 x^3+115 x^2+494 x+100\right )}{(5 x-1) \left (x^2+4\right ) \left (75 \log \left (x^2+4\right ) x^2+225 x^2-30 \log \left (x^2+4\right ) x-115 x+3 \log \left (x^2+4\right )+9\right )}\right )dx\) |
Input:
Int[(50000 + 247000*x + 57500*x^2 - 162500*x^3 + 375000*x^4 + (30000 + 148 200*x + 34500*x^2 - 97500*x^3 + 225000*x^4)*Log[(25*x + (-3 + 30*x - 75*x^ 2)*Log[E^3*(4 + x^2)])/(3 - 30*x + 75*x^2)] + (6000 + 29640*x + 6900*x^2 - 19500*x^3 + 45000*x^4)*Log[(25*x + (-3 + 30*x - 75*x^2)*Log[E^3*(4 + x^2) ])/(3 - 30*x + 75*x^2)]^2 + (400 + 1976*x + 460*x^2 - 1300*x^3 + 3000*x^4) *Log[(25*x + (-3 + 30*x - 75*x^2)*Log[E^3*(4 + x^2)])/(3 - 30*x + 75*x^2)] ^3)/(100*x - 500*x^2 + 25*x^3 - 125*x^4 + (-12 + 180*x - 903*x^2 + 1545*x^ 3 - 225*x^4 + 375*x^5)*Log[E^3*(4 + x^2)]),x]
Output:
$Aborted
\[\int \frac {\left (3000 x^{4}-1300 x^{3}+460 x^{2}+1976 x +400\right ) {\ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )}^{3}+\left (45000 x^{4}-19500 x^{3}+6900 x^{2}+29640 x +6000\right ) {\ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )}^{2}+\left (225000 x^{4}-97500 x^{3}+34500 x^{2}+148200 x +30000\right ) \ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )+375000 x^{4}-162500 x^{3}+57500 x^{2}+247000 x +50000}{\left (375 x^{5}-225 x^{4}+1545 x^{3}-903 x^{2}+180 x -12\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )-125 x^{4}+25 x^{3}-500 x^{2}+100 x}d x\]
Input:
int(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*ln(((-75*x^2+30*x-3)*ln((x^2+4 )*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+29640*x+6 000)*ln(((-75*x^2+30*x-3)*ln((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^2+(225 000*x^4-97500*x^3+34500*x^2+148200*x+30000)*ln(((-75*x^2+30*x-3)*ln((x^2+4 )*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x^2+247000*x+ 50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*ln((x^2+4)*exp(3))-125 *x^4+25*x^3-500*x^2+100*x),x)
Output:
int(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*ln(((-75*x^2+30*x-3)*ln((x^2+4 )*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+29640*x+6 000)*ln(((-75*x^2+30*x-3)*ln((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^2+(225 000*x^4-97500*x^3+34500*x^2+148200*x+30000)*ln(((-75*x^2+30*x-3)*ln((x^2+4 )*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x^2+247000*x+ 50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*ln((x^2+4)*exp(3))-125 *x^4+25*x^3-500*x^2+100*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.77 \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{4} + 20 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} + 150 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 500 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) \] Input:
integrate(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*log(((-75*x^2+30*x-3)*lo g((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+2 9640*x+6000)*log(((-75*x^2+30*x-3)*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+ 3))^2+(225000*x^4-97500*x^3+34500*x^2+148200*x+30000)*log(((-75*x^2+30*x-3 )*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x ^2+247000*x+50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*log((x^2+4 )*exp(3))-125*x^4+25*x^3-500*x^2+100*x),x, algorithm="fricas")
Output:
log(-1/3*(3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1))^4 + 20*log(-1/3*(3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(2 5*x^2 - 10*x + 1))^3 + 150*log(-1/3*(3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e ^3) - 25*x)/(25*x^2 - 10*x + 1))^2 + 500*log(-1/3*(3*(25*x^2 - 10*x + 1)*l og((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (24) = 48\).
Time = 0.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.70 \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{4} + 20 \log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{3} + 150 \log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{2} + 500 \log {\left (- \frac {25 x}{75 x^{2} - 30 x + 3} + \log {\left (\left (x^{2} + 4\right ) e^{3} \right )} \right )} \] Input:
integrate(((3000*x**4-1300*x**3+460*x**2+1976*x+400)*ln(((-75*x**2+30*x-3) *ln((x**2+4)*exp(3))+25*x)/(75*x**2-30*x+3))**3+(45000*x**4-19500*x**3+690 0*x**2+29640*x+6000)*ln(((-75*x**2+30*x-3)*ln((x**2+4)*exp(3))+25*x)/(75*x **2-30*x+3))**2+(225000*x**4-97500*x**3+34500*x**2+148200*x+30000)*ln(((-7 5*x**2+30*x-3)*ln((x**2+4)*exp(3))+25*x)/(75*x**2-30*x+3))+375000*x**4-162 500*x**3+57500*x**2+247000*x+50000)/((375*x**5-225*x**4+1545*x**3-903*x**2 +180*x-12)*ln((x**2+4)*exp(3))-125*x**4+25*x**3-500*x**2+100*x),x)
Output:
log((25*x + (-75*x**2 + 30*x - 3)*log((x**2 + 4)*exp(3)))/(75*x**2 - 30*x + 3))**4 + 20*log((25*x + (-75*x**2 + 30*x - 3)*log((x**2 + 4)*exp(3)))/(7 5*x**2 - 30*x + 3))**3 + 150*log((25*x + (-75*x**2 + 30*x - 3)*log((x**2 + 4)*exp(3)))/(75*x**2 - 30*x + 3))**2 + 500*log(-25*x/(75*x**2 - 30*x + 3) + log((x**2 + 4)*exp(3)))
\[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\int { -\frac {4 \, {\left (93750 \, x^{4} + {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} - 40625 \, x^{3} + 15 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 14375 \, x^{2} + 75 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) + 61750 \, x + 12500\right )}}{125 \, x^{4} - 25 \, x^{3} + 500 \, x^{2} - 3 \, {\left (125 \, x^{5} - 75 \, x^{4} + 515 \, x^{3} - 301 \, x^{2} + 60 \, x - 4\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 100 \, x} \,d x } \] Input:
integrate(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*log(((-75*x^2+30*x-3)*lo g((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+2 9640*x+6000)*log(((-75*x^2+30*x-3)*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+ 3))^2+(225000*x^4-97500*x^3+34500*x^2+148200*x+30000)*log(((-75*x^2+30*x-3 )*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x ^2+247000*x+50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*log((x^2+4 )*exp(3))-125*x^4+25*x^3-500*x^2+100*x),x, algorithm="maxima")
Output:
-4*integrate((93750*x^4 + (750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log( -1/3*(3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1) )^3 - 40625*x^3 + 15*(750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log(-1/3* (3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1))^2 + 14375*x^2 + 75*(750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log(-1/3*(3*(2 5*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1)) + 61750* x + 12500)/(125*x^4 - 25*x^3 + 500*x^2 - 3*(125*x^5 - 75*x^4 + 515*x^3 - 3 01*x^2 + 60*x - 4)*log((x^2 + 4)*e^3) - 100*x), x)
\[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\int { -\frac {4 \, {\left (93750 \, x^{4} + {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} - 40625 \, x^{3} + 15 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 14375 \, x^{2} + 75 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) + 61750 \, x + 12500\right )}}{125 \, x^{4} - 25 \, x^{3} + 500 \, x^{2} - 3 \, {\left (125 \, x^{5} - 75 \, x^{4} + 515 \, x^{3} - 301 \, x^{2} + 60 \, x - 4\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 100 \, x} \,d x } \] Input:
integrate(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*log(((-75*x^2+30*x-3)*lo g((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+2 9640*x+6000)*log(((-75*x^2+30*x-3)*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+ 3))^2+(225000*x^4-97500*x^3+34500*x^2+148200*x+30000)*log(((-75*x^2+30*x-3 )*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x ^2+247000*x+50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*log((x^2+4 )*exp(3))-125*x^4+25*x^3-500*x^2+100*x),x, algorithm="giac")
Output:
integrate(-4*(93750*x^4 + (750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log( -1/3*(3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1) )^3 - 40625*x^3 + 15*(750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log(-1/3* (3*(25*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1))^2 + 14375*x^2 + 75*(750*x^4 - 325*x^3 + 115*x^2 + 494*x + 100)*log(-1/3*(3*(2 5*x^2 - 10*x + 1)*log((x^2 + 4)*e^3) - 25*x)/(25*x^2 - 10*x + 1)) + 61750* x + 12500)/(125*x^4 - 25*x^3 + 500*x^2 - 3*(125*x^5 - 75*x^4 + 515*x^3 - 3 01*x^2 + 60*x - 4)*log((x^2 + 4)*e^3) - 100*x), x)
Timed out. \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=\int \frac {247000\,x+\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )\,\left (225000\,x^4-97500\,x^3+34500\,x^2+148200\,x+30000\right )+{\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )}^3\,\left (3000\,x^4-1300\,x^3+460\,x^2+1976\,x+400\right )+{\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )}^2\,\left (45000\,x^4-19500\,x^3+6900\,x^2+29640\,x+6000\right )+57500\,x^2-162500\,x^3+375000\,x^4+50000}{100\,x-500\,x^2+25\,x^3-125\,x^4+\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (375\,x^5-225\,x^4+1545\,x^3-903\,x^2+180\,x-12\right )} \,d x \] Input:
int((247000*x + log((25*x - log(exp(3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75 *x^2 - 30*x + 3))*(148200*x + 34500*x^2 - 97500*x^3 + 225000*x^4 + 30000) + log((25*x - log(exp(3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75*x^2 - 30*x + 3))^3*(1976*x + 460*x^2 - 1300*x^3 + 3000*x^4 + 400) + log((25*x - log(exp (3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75*x^2 - 30*x + 3))^2*(29640*x + 6900 *x^2 - 19500*x^3 + 45000*x^4 + 6000) + 57500*x^2 - 162500*x^3 + 375000*x^4 + 50000)/(100*x - 500*x^2 + 25*x^3 - 125*x^4 + log(exp(3)*(x^2 + 4))*(180 *x - 903*x^2 + 1545*x^3 - 225*x^4 + 375*x^5 - 12)),x)
Output:
int((247000*x + log((25*x - log(exp(3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75 *x^2 - 30*x + 3))*(148200*x + 34500*x^2 - 97500*x^3 + 225000*x^4 + 30000) + log((25*x - log(exp(3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75*x^2 - 30*x + 3))^3*(1976*x + 460*x^2 - 1300*x^3 + 3000*x^4 + 400) + log((25*x - log(exp (3)*(x^2 + 4))*(75*x^2 - 30*x + 3))/(75*x^2 - 30*x + 3))^2*(29640*x + 6900 *x^2 - 19500*x^3 + 45000*x^4 + 6000) + 57500*x^2 - 162500*x^3 + 375000*x^4 + 50000)/(100*x - 500*x^2 + 25*x^3 - 125*x^4 + log(exp(3)*(x^2 + 4))*(180 *x - 903*x^2 + 1545*x^3 - 225*x^4 + 375*x^5 - 12)), x)
Time = 0.19 (sec) , antiderivative size = 288, normalized size of antiderivative = 9.60 \[ \int \frac {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx=500 \,\mathrm {log}\left (75 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x^{2}-30 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x +3 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right )-25 x \right )-1000 \,\mathrm {log}\left (5 x -1\right )+{\mathrm {log}\left (\frac {-75 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x^{2}+30 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x -3 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )}^{4}+20 {\mathrm {log}\left (\frac {-75 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x^{2}+30 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x -3 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )}^{3}+150 {\mathrm {log}\left (\frac {-75 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x^{2}+30 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right ) x -3 \,\mathrm {log}\left (e^{3} x^{2}+4 e^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )}^{2} \] Input:
int(((3000*x^4-1300*x^3+460*x^2+1976*x+400)*log(((-75*x^2+30*x-3)*log((x^2 +4)*exp(3))+25*x)/(75*x^2-30*x+3))^3+(45000*x^4-19500*x^3+6900*x^2+29640*x +6000)*log(((-75*x^2+30*x-3)*log((x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))^2+ (225000*x^4-97500*x^3+34500*x^2+148200*x+30000)*log(((-75*x^2+30*x-3)*log( (x^2+4)*exp(3))+25*x)/(75*x^2-30*x+3))+375000*x^4-162500*x^3+57500*x^2+247 000*x+50000)/((375*x^5-225*x^4+1545*x^3-903*x^2+180*x-12)*log((x^2+4)*exp( 3))-125*x^4+25*x^3-500*x^2+100*x),x)
Output:
500*log(75*log(e**3*x**2 + 4*e**3)*x**2 - 30*log(e**3*x**2 + 4*e**3)*x + 3 *log(e**3*x**2 + 4*e**3) - 25*x) - 1000*log(5*x - 1) + log(( - 75*log(e**3 *x**2 + 4*e**3)*x**2 + 30*log(e**3*x**2 + 4*e**3)*x - 3*log(e**3*x**2 + 4* e**3) + 25*x)/(75*x**2 - 30*x + 3))**4 + 20*log(( - 75*log(e**3*x**2 + 4*e **3)*x**2 + 30*log(e**3*x**2 + 4*e**3)*x - 3*log(e**3*x**2 + 4*e**3) + 25* x)/(75*x**2 - 30*x + 3))**3 + 150*log(( - 75*log(e**3*x**2 + 4*e**3)*x**2 + 30*log(e**3*x**2 + 4*e**3)*x - 3*log(e**3*x**2 + 4*e**3) + 25*x)/(75*x** 2 - 30*x + 3))**2