Integrand size = 253, antiderivative size = 34 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=1+\frac {4+x}{2 (-4+x)+\frac {x}{\left (5+3 x^2 \left (e^x+x\right )\right ) \log (x)}} \] Output:
1+(4+x)/(x/(3*x^2*(exp(x)+x)+5)/ln(x)+2*x-8)
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=-\frac {x-16 \left (5+3 e^x x^2+3 x^3\right ) \log (x)}{2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \] Input:
Integrate[(20 + 5*x + 12*x^3 + 3*x^4 + E^x*(12*x^2 + 3*x^3) + (-20 + 24*x^ 3 + 9*x^4 + E^x*(12*x^2 + 18*x^3 + 3*x^4))*Log[x] + (-400 - 480*x^3 - 144* E^(2*x)*x^4 - 144*x^6 + E^x*(-480*x^2 - 288*x^5))*Log[x]^2)/(x^2 + (-80*x + 20*x^2 - 48*x^4 + 12*x^5 + E^x*(-48*x^3 + 12*x^4))*Log[x] + (1600 - 800* x + 100*x^2 + 1920*x^3 - 960*x^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + E^(2*x)*(576*x^4 - 288*x^5 + 36*x^6) + E^x*(1920*x^2 - 960*x^3 + 120*x^4 + 1152*x^5 - 576*x^6 + 72*x^7))*Log[x]^2),x]
Output:
-1/2*(x - 16*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])/(x + 2*(-4 + x)*(5 + 3*E^x*x^ 2 + 3*x^3)*Log[x])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {3 x^4+12 x^3+e^x \left (3 x^3+12 x^2\right )+\left (9 x^4+24 x^3+e^x \left (3 x^4+18 x^3+12 x^2\right )-20\right ) \log (x)+\left (-144 x^6-144 e^{2 x} x^4-480 x^3+e^x \left (-288 x^5-480 x^2\right )-400\right ) \log ^2(x)+5 x+20}{x^2+\left (12 x^5-48 x^4+20 x^2+e^x \left (12 x^4-48 x^3\right )-80 x\right ) \log (x)+\left (36 x^8-288 x^7+576 x^6+120 x^5-960 x^4+1920 x^3+100 x^2+e^{2 x} \left (36 x^6-288 x^5+576 x^4\right )+e^x \left (72 x^7-576 x^6+1152 x^5+120 x^4-960 x^3+1920 x^2\right )-800 x+1600\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+4) \left (3 x^3+3 e^x x^2+5\right )-16 \left (3 x^3+3 e^x x^2+5\right )^2 \log ^2(x)+\left (9 x^4+24 x^3+3 e^x \left (x^2+6 x+4\right ) x^2-20\right ) \log (x)}{\left (2 (x-4) \left (3 x^3+3 e^x x^2+5\right ) \log (x)+x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3 \log (x)+x^2+2 x^2 \log (x)-4 x \log (x)-16 \log (x)-16}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )}-\frac {(x+4) \left (6 x^6 \log ^2(x)-54 x^5 \log ^2(x)+144 x^4 \log ^2(x)-86 x^3 \log ^2(x)+x^3 \log (x)+x^2-60 x^2 \log ^2(x)-2 x^2 \log (x)-4 x+320 \log ^2(x)-4 x \log (x)\right )}{2 (x-4)^2 \log (x) \left (6 x^4 \log (x)+6 e^x x^3 \log (x)-24 x^3 \log (x)-24 e^x x^2 \log (x)+x+10 x \log (x)-40 \log (x)\right )^2}-\frac {4}{(x-4)^2}\right )dx\) |
Input:
Int[(20 + 5*x + 12*x^3 + 3*x^4 + E^x*(12*x^2 + 3*x^3) + (-20 + 24*x^3 + 9* x^4 + E^x*(12*x^2 + 18*x^3 + 3*x^4))*Log[x] + (-400 - 480*x^3 - 144*E^(2*x )*x^4 - 144*x^6 + E^x*(-480*x^2 - 288*x^5))*Log[x]^2)/(x^2 + (-80*x + 20*x ^2 - 48*x^4 + 12*x^5 + E^x*(-48*x^3 + 12*x^4))*Log[x] + (1600 - 800*x + 10 0*x^2 + 1920*x^3 - 960*x^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + E^(2*x )*(576*x^4 - 288*x^5 + 36*x^6) + E^x*(1920*x^2 - 960*x^3 + 120*x^4 + 1152* x^5 - 576*x^6 + 72*x^7))*Log[x]^2),x]
Output:
$Aborted
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91
\[\frac {4}{x -4}-\frac {\left (4+x \right ) x}{2 \left (x -4\right ) \left (6 x^{3} {\mathrm e}^{x} \ln \left (x \right )+6 x^{4} \ln \left (x \right )-24 x^{2} {\mathrm e}^{x} \ln \left (x \right )-24 x^{3} \ln \left (x \right )+10 x \ln \left (x \right )-40 \ln \left (x \right )+x \right )}\]
Input:
int(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*ln( x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9*x^4+24*x^3-20)*ln(x)+(3*x^3+12*x^2)*e xp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72*x^7-576 *x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+120* x^5-960*x^4+1920*x^3+100*x^2-800*x+1600)*ln(x)^2+((12*x^4-48*x^3)*exp(x)+1 2*x^5-48*x^4+20*x^2-80*x)*ln(x)+x^2),x)
Output:
4/(x-4)-1/2*(4+x)*x/(x-4)/(6*x^3*exp(x)*ln(x)+6*x^4*ln(x)-24*x^2*exp(x)*ln (x)-24*x^3*ln(x)+10*x*ln(x)-40*ln(x)+x)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.76 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\frac {16 \, {\left (3 \, x^{3} + 3 \, x^{2} e^{x} + 5\right )} \log \left (x\right ) - x}{2 \, {\left (2 \, {\left (3 \, x^{4} - 12 \, x^{3} + 3 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{x} + 5 \, x - 20\right )} \log \left (x\right ) + x\right )}} \] Input:
integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-40 0)*log(x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+1 2*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72 *x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576* x^6+120*x^5-960*x^4+1920*x^3+100*x^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3) *exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="fricas")
Output:
1/2*(16*(3*x^3 + 3*x^2*e^x + 5)*log(x) - x)/(2*(3*x^4 - 12*x^3 + 3*(x^3 - 4*x^2)*e^x + 5*x - 20)*log(x) + x)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (26) = 52\).
Time = 0.54 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.79 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\frac {- x^{2} - 4 x}{12 x^{5} \log {\left (x \right )} - 96 x^{4} \log {\left (x \right )} + 192 x^{3} \log {\left (x \right )} + 20 x^{2} \log {\left (x \right )} + 2 x^{2} - 160 x \log {\left (x \right )} - 8 x + \left (12 x^{4} \log {\left (x \right )} - 96 x^{3} \log {\left (x \right )} + 192 x^{2} \log {\left (x \right )}\right ) e^{x} + 320 \log {\left (x \right )}} + \frac {4}{x - 4} \] Input:
integrate(((-144*exp(x)**2*x**4+(-288*x**5-480*x**2)*exp(x)-144*x**6-480*x **3-400)*ln(x)**2+((3*x**4+18*x**3+12*x**2)*exp(x)+9*x**4+24*x**3-20)*ln(x )+(3*x**3+12*x**2)*exp(x)+3*x**4+12*x**3+5*x+20)/(((36*x**6-288*x**5+576*x **4)*exp(x)**2+(72*x**7-576*x**6+1152*x**5+120*x**4-960*x**3+1920*x**2)*ex p(x)+36*x**8-288*x**7+576*x**6+120*x**5-960*x**4+1920*x**3+100*x**2-800*x+ 1600)*ln(x)**2+((12*x**4-48*x**3)*exp(x)+12*x**5-48*x**4+20*x**2-80*x)*ln( x)+x**2),x)
Output:
(-x**2 - 4*x)/(12*x**5*log(x) - 96*x**4*log(x) + 192*x**3*log(x) + 20*x**2 *log(x) + 2*x**2 - 160*x*log(x) - 8*x + (12*x**4*log(x) - 96*x**3*log(x) + 192*x**2*log(x))*exp(x) + 320*log(x)) + 4/(x - 4)
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.88 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\frac {48 \, x^{2} e^{x} \log \left (x\right ) + 16 \, {\left (3 \, x^{3} + 5\right )} \log \left (x\right ) - x}{2 \, {\left (6 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{x} \log \left (x\right ) + 2 \, {\left (3 \, x^{4} - 12 \, x^{3} + 5 \, x - 20\right )} \log \left (x\right ) + x\right )}} \] Input:
integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-40 0)*log(x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+1 2*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72 *x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576* x^6+120*x^5-960*x^4+1920*x^3+100*x^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3) *exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="maxima")
Output:
1/2*(48*x^2*e^x*log(x) + 16*(3*x^3 + 5)*log(x) - x)/(6*(x^3 - 4*x^2)*e^x*l og(x) + 2*(3*x^4 - 12*x^3 + 5*x - 20)*log(x) + x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (32) = 64\).
Time = 0.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.09 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\frac {48 \, x^{3} \log \left (x\right ) + 48 \, x^{2} e^{x} \log \left (x\right ) - x + 80 \, \log \left (x\right )}{2 \, {\left (6 \, x^{4} \log \left (x\right ) + 6 \, x^{3} e^{x} \log \left (x\right ) - 24 \, x^{3} \log \left (x\right ) - 24 \, x^{2} e^{x} \log \left (x\right ) + 10 \, x \log \left (x\right ) + x - 40 \, \log \left (x\right )\right )}} \] Input:
integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-40 0)*log(x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+1 2*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72 *x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576* x^6+120*x^5-960*x^4+1920*x^3+100*x^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3) *exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="giac")
Output:
1/2*(48*x^3*log(x) + 48*x^2*e^x*log(x) - x + 80*log(x))/(6*x^4*log(x) + 6* x^3*e^x*log(x) - 24*x^3*log(x) - 24*x^2*e^x*log(x) + 10*x*log(x) + x - 40* log(x))
Timed out. \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\int \frac {5\,x+{\mathrm {e}}^x\,\left (3\,x^3+12\,x^2\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (3\,x^4+18\,x^3+12\,x^2\right )+24\,x^3+9\,x^4-20\right )+12\,x^3+3\,x^4-{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (288\,x^5+480\,x^2\right )+144\,x^4\,{\mathrm {e}}^{2\,x}+480\,x^3+144\,x^6+400\right )+20}{x^2-\ln \left (x\right )\,\left (80\,x+{\mathrm {e}}^x\,\left (48\,x^3-12\,x^4\right )-20\,x^2+48\,x^4-12\,x^5\right )+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (36\,x^6-288\,x^5+576\,x^4\right )-800\,x+100\,x^2+1920\,x^3-960\,x^4+120\,x^5+576\,x^6-288\,x^7+36\,x^8+{\mathrm {e}}^x\,\left (72\,x^7-576\,x^6+1152\,x^5+120\,x^4-960\,x^3+1920\,x^2\right )+1600\right )} \,d x \] Input:
int((5*x + exp(x)*(12*x^2 + 3*x^3) + log(x)*(exp(x)*(12*x^2 + 18*x^3 + 3*x ^4) + 24*x^3 + 9*x^4 - 20) + 12*x^3 + 3*x^4 - log(x)^2*(exp(x)*(480*x^2 + 288*x^5) + 144*x^4*exp(2*x) + 480*x^3 + 144*x^6 + 400) + 20)/(x^2 - log(x) *(80*x + exp(x)*(48*x^3 - 12*x^4) - 20*x^2 + 48*x^4 - 12*x^5) + log(x)^2*( exp(2*x)*(576*x^4 - 288*x^5 + 36*x^6) - 800*x + 100*x^2 + 1920*x^3 - 960*x ^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + exp(x)*(1920*x^2 - 960*x^3 + 1 20*x^4 + 1152*x^5 - 576*x^6 + 72*x^7) + 1600)),x)
Output:
int((5*x + exp(x)*(12*x^2 + 3*x^3) + log(x)*(exp(x)*(12*x^2 + 18*x^3 + 3*x ^4) + 24*x^3 + 9*x^4 - 20) + 12*x^3 + 3*x^4 - log(x)^2*(exp(x)*(480*x^2 + 288*x^5) + 144*x^4*exp(2*x) + 480*x^3 + 144*x^6 + 400) + 20)/(x^2 - log(x) *(80*x + exp(x)*(48*x^3 - 12*x^4) - 20*x^2 + 48*x^4 - 12*x^5) + log(x)^2*( exp(2*x)*(576*x^4 - 288*x^5 + 36*x^6) - 800*x + 100*x^2 + 1920*x^3 - 960*x ^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + exp(x)*(1920*x^2 - 960*x^3 + 1 20*x^4 + 1152*x^5 - 576*x^6 + 72*x^7) + 1600)), x)
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.38 \[ \int \frac {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx=\frac {\mathrm {log}\left (x \right ) \left (3 e^{x} x^{3}+12 e^{x} x^{2}+3 x^{4}+12 x^{3}+5 x +20\right )}{6 e^{x} \mathrm {log}\left (x \right ) x^{3}-24 e^{x} \mathrm {log}\left (x \right ) x^{2}+6 \,\mathrm {log}\left (x \right ) x^{4}-24 \,\mathrm {log}\left (x \right ) x^{3}+10 \,\mathrm {log}\left (x \right ) x -40 \,\mathrm {log}\left (x \right )+x} \] Input:
int(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*log (x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+12*x^2) *exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72*x^7-5 76*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+12 0*x^5-960*x^4+1920*x^3+100*x^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3)*exp(x )+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x)
Output:
(log(x)*(3*e**x*x**3 + 12*e**x*x**2 + 3*x**4 + 12*x**3 + 5*x + 20))/(6*e** x*log(x)*x**3 - 24*e**x*log(x)*x**2 + 6*log(x)*x**4 - 24*log(x)*x**3 + 10* log(x)*x - 40*log(x) + x)