Integrand size = 277, antiderivative size = 30 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {\log (3+\log (3-x))}{x \left (2+x^2 \left (-3 e^{-x}+x\right )\right )} \] Output:
ln(ln(3-x)+3)/((x-3/exp(x))*x^2+2)/x
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {e^x \log (3+\log (3-x))}{-3 x^3+e^x x \left (2+x^3\right )} \] Input:
Integrate[(-3*E^x*x^3 + E^(2*x)*(2*x + x^4) + (E^(2*x)*(18 - 6*x + 36*x^3 - 12*x^4) + E^x*(-81*x^2 + 54*x^3 - 9*x^4) + (E^(2*x)*(6 - 2*x + 12*x^3 - 4*x^4) + E^x*(-27*x^2 + 18*x^3 - 3*x^4))*Log[3 - x])*Log[3 + Log[3 - x]])/ (-81*x^6 + 27*x^7 + E^x*(108*x^4 - 36*x^5 + 54*x^7 - 18*x^8) + E^(2*x)*(-3 6*x^2 + 12*x^3 - 36*x^5 + 12*x^6 - 9*x^8 + 3*x^9) + (-27*x^6 + 9*x^7 + E^x *(36*x^4 - 12*x^5 + 18*x^7 - 6*x^8) + E^(2*x)*(-12*x^2 + 4*x^3 - 12*x^5 + 4*x^6 - 3*x^8 + x^9))*Log[3 - x]),x]
Output:
(E^x*Log[3 + Log[3 - x]])/(-3*x^3 + E^x*x*(2 + x^3))
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.
| \(\displaystyle \int \frac {e^{2 x} \left (x^4+2 x\right )-3 e^x x^3+\left (e^{2 x} \left (-12 x^4+36 x^3-6 x+18\right )+e^x \left (-9 x^4+54 x^3-81 x^2\right )+\left (e^{2 x} \left (-4 x^4+12 x^3-2 x+6\right )+e^x \left (-3 x^4+18 x^3-27 x^2\right )\right ) \log (3-x)\right ) \log (\log (3-x)+3)}{27 x^7-81 x^6+e^x \left (-18 x^8+54 x^7-36 x^5+108 x^4\right )+e^{2 x} \left (3 x^9-9 x^8+12 x^6-36 x^5+12 x^3-36 x^2\right )+\left (9 x^7-27 x^6+e^x \left (-6 x^8+18 x^7-12 x^5+36 x^4\right )+e^{2 x} \left (x^9-3 x^8+4 x^6-12 x^5+4 x^3-12 x^2\right )\right ) \log (3-x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (3 x^3-e^x \left (x^3+2\right ) x+(x-3) \left (e^x \left (4 x^3+2\right )+3 (x-3) x^2\right ) (\log (3-x)+3) \log (\log (3-x)+3)\right )}{(3-x) \left (3 x^3-e^x x \left (x^3+2\right )\right )^2 (\log (3-x)+3)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^x \left (x^4+x^3+2 x-4\right ) \log (\log (3-x)+3)}{\left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right )^2}-\frac {e^x \left (-x^4+4 x^4 \log (3-x) \log (\log (3-x)+3)+12 x^4 \log (\log (3-x)+3)-12 x^3 \log (3-x) \log (\log (3-x)+3)-36 x^3 \log (\log (3-x)+3)-2 x+2 x \log (3-x) \log (\log (3-x)+3)+6 x \log (\log (3-x)+3)-6 \log (3-x) \log (\log (3-x)+3)-18 \log (\log (3-x)+3)\right )}{(x-3) x^2 \left (x^3+2\right ) \left (e^x x^3-3 x^2+2 e^x\right ) (\log (3-x)+3)}\right )dx\) |
Input:
Int[(-3*E^x*x^3 + E^(2*x)*(2*x + x^4) + (E^(2*x)*(18 - 6*x + 36*x^3 - 12*x ^4) + E^x*(-81*x^2 + 54*x^3 - 9*x^4) + (E^(2*x)*(6 - 2*x + 12*x^3 - 4*x^4) + E^x*(-27*x^2 + 18*x^3 - 3*x^4))*Log[3 - x])*Log[3 + Log[3 - x]])/(-81*x ^6 + 27*x^7 + E^x*(108*x^4 - 36*x^5 + 54*x^7 - 18*x^8) + E^(2*x)*(-36*x^2 + 12*x^3 - 36*x^5 + 12*x^6 - 9*x^8 + 3*x^9) + (-27*x^6 + 9*x^7 + E^x*(36*x ^4 - 12*x^5 + 18*x^7 - 6*x^8) + E^(2*x)*(-12*x^2 + 4*x^3 - 12*x^5 + 4*x^6 - 3*x^8 + x^9))*Log[3 - x]),x]
Output:
$Aborted
Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
\[\frac {{\mathrm e}^{x} \ln \left (\ln \left (-x +3\right )+3\right )}{x \left ({\mathrm e}^{x} x^{3}-3 x^{2}+2 \,{\mathrm e}^{x}\right )}\]
Input:
int(((((-4*x^4+12*x^3-2*x+6)*exp(x)^2+(-3*x^4+18*x^3-27*x^2)*exp(x))*ln(-x +3)+(-12*x^4+36*x^3-6*x+18)*exp(x)^2+(-9*x^4+54*x^3-81*x^2)*exp(x))*ln(ln( -x+3)+3)+(x^4+2*x)*exp(x)^2-3*exp(x)*x^3)/(((x^9-3*x^8+4*x^6-12*x^5+4*x^3- 12*x^2)*exp(x)^2+(-6*x^8+18*x^7-12*x^5+36*x^4)*exp(x)+9*x^7-27*x^6)*ln(-x+ 3)+(3*x^9-9*x^8+12*x^6-36*x^5+12*x^3-36*x^2)*exp(x)^2+(-18*x^8+54*x^7-36*x ^5+108*x^4)*exp(x)+27*x^7-81*x^6),x)
Output:
exp(x)/x/(exp(x)*x^3-3*x^2+2*exp(x))*ln(ln(-x+3)+3)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=-\frac {e^{x} \log \left (\log \left (-x + 3\right ) + 3\right )}{3 \, x^{3} - {\left (x^{4} + 2 \, x\right )} e^{x}} \] Input:
integrate(((((-4*x^4+12*x^3-2*x+6)*exp(x)^2+(-3*x^4+18*x^3-27*x^2)*exp(x)) *log(3-x)+(-12*x^4+36*x^3-6*x+18)*exp(x)^2+(-9*x^4+54*x^3-81*x^2)*exp(x))* log(log(3-x)+3)+(x^4+2*x)*exp(x)^2-3*exp(x)*x^3)/(((x^9-3*x^8+4*x^6-12*x^5 +4*x^3-12*x^2)*exp(x)^2+(-6*x^8+18*x^7-12*x^5+36*x^4)*exp(x)+9*x^7-27*x^6) *log(3-x)+(3*x^9-9*x^8+12*x^6-36*x^5+12*x^3-36*x^2)*exp(x)^2+(-18*x^8+54*x ^7-36*x^5+108*x^4)*exp(x)+27*x^7-81*x^6),x, algorithm="fricas")
Output:
-e^x*log(log(-x + 3) + 3)/(3*x^3 - (x^4 + 2*x)*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {3 x \log {\left (\log {\left (3 - x \right )} + 3 \right )}}{- 3 x^{5} - 6 x^{2} + \left (x^{6} + 4 x^{3} + 4\right ) e^{x}} + \frac {\log {\left (\log {\left (3 - x \right )} + 3 \right )}}{x^{4} + 2 x} \] Input:
integrate(((((-4*x**4+12*x**3-2*x+6)*exp(x)**2+(-3*x**4+18*x**3-27*x**2)*e xp(x))*ln(3-x)+(-12*x**4+36*x**3-6*x+18)*exp(x)**2+(-9*x**4+54*x**3-81*x** 2)*exp(x))*ln(ln(3-x)+3)+(x**4+2*x)*exp(x)**2-3*exp(x)*x**3)/(((x**9-3*x** 8+4*x**6-12*x**5+4*x**3-12*x**2)*exp(x)**2+(-6*x**8+18*x**7-12*x**5+36*x** 4)*exp(x)+9*x**7-27*x**6)*ln(3-x)+(3*x**9-9*x**8+12*x**6-36*x**5+12*x**3-3 6*x**2)*exp(x)**2+(-18*x**8+54*x**7-36*x**5+108*x**4)*exp(x)+27*x**7-81*x* *6),x)
Output:
3*x*log(log(3 - x) + 3)/(-3*x**5 - 6*x**2 + (x**6 + 4*x**3 + 4)*exp(x)) + log(log(3 - x) + 3)/(x**4 + 2*x)
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=-\frac {e^{x} \log \left (\log \left (-x + 3\right ) + 3\right )}{3 \, x^{3} - {\left (x^{4} + 2 \, x\right )} e^{x}} \] Input:
integrate(((((-4*x^4+12*x^3-2*x+6)*exp(x)^2+(-3*x^4+18*x^3-27*x^2)*exp(x)) *log(3-x)+(-12*x^4+36*x^3-6*x+18)*exp(x)^2+(-9*x^4+54*x^3-81*x^2)*exp(x))* log(log(3-x)+3)+(x^4+2*x)*exp(x)^2-3*exp(x)*x^3)/(((x^9-3*x^8+4*x^6-12*x^5 +4*x^3-12*x^2)*exp(x)^2+(-6*x^8+18*x^7-12*x^5+36*x^4)*exp(x)+9*x^7-27*x^6) *log(3-x)+(3*x^9-9*x^8+12*x^6-36*x^5+12*x^3-36*x^2)*exp(x)^2+(-18*x^8+54*x ^7-36*x^5+108*x^4)*exp(x)+27*x^7-81*x^6),x, algorithm="maxima")
Output:
-e^x*log(log(-x + 3) + 3)/(3*x^3 - (x^4 + 2*x)*e^x)
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {e^{x} \log \left (\log \left (-x + 3\right ) + 3\right )}{x^{4} e^{x} - 3 \, x^{3} + 2 \, x e^{x}} \] Input:
integrate(((((-4*x^4+12*x^3-2*x+6)*exp(x)^2+(-3*x^4+18*x^3-27*x^2)*exp(x)) *log(3-x)+(-12*x^4+36*x^3-6*x+18)*exp(x)^2+(-9*x^4+54*x^3-81*x^2)*exp(x))* log(log(3-x)+3)+(x^4+2*x)*exp(x)^2-3*exp(x)*x^3)/(((x^9-3*x^8+4*x^6-12*x^5 +4*x^3-12*x^2)*exp(x)^2+(-6*x^8+18*x^7-12*x^5+36*x^4)*exp(x)+9*x^7-27*x^6) *log(3-x)+(3*x^9-9*x^8+12*x^6-36*x^5+12*x^3-36*x^2)*exp(x)^2+(-18*x^8+54*x ^7-36*x^5+108*x^4)*exp(x)+27*x^7-81*x^6),x, algorithm="giac")
Output:
e^x*log(log(-x + 3) + 3)/(x^4*e^x - 3*x^3 + 2*x*e^x)
Time = 3.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.70 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {3\,\ln \left (\ln \left (3-x\right )+3\right )\,\left (3\,x^3-x^4\right )}{\left (x^3+2\right )\,\left ({\mathrm {e}}^x\,\left (-x^6+3\,x^5-2\,x^3+6\,x^2\right )-9\,x^4+3\,x^5\right )}-\frac {\ln \left (\ln \left (3-x\right )+3\right )\,\left (\frac {9\,x^4-3\,x^5}{x^4+2\,x}-\frac {{\mathrm {e}}^x\,\left (-x^6+3\,x^5-2\,x^3+6\,x^2\right )}{x^4+2\,x}\right )}{{\mathrm {e}}^x\,\left (-x^6+3\,x^5-2\,x^3+6\,x^2\right )-9\,x^4+3\,x^5} \] Input:
int((3*x^3*exp(x) + log(log(3 - x) + 3)*(exp(2*x)*(6*x - 36*x^3 + 12*x^4 - 18) + exp(x)*(81*x^2 - 54*x^3 + 9*x^4) + log(3 - x)*(exp(2*x)*(2*x - 12*x ^3 + 4*x^4 - 6) + exp(x)*(27*x^2 - 18*x^3 + 3*x^4))) - exp(2*x)*(2*x + x^4 ))/(exp(2*x)*(36*x^2 - 12*x^3 + 36*x^5 - 12*x^6 + 9*x^8 - 3*x^9) - exp(x)* (108*x^4 - 36*x^5 + 54*x^7 - 18*x^8) + log(3 - x)*(exp(2*x)*(12*x^2 - 4*x^ 3 + 12*x^5 - 4*x^6 + 3*x^8 - x^9) - exp(x)*(36*x^4 - 12*x^5 + 18*x^7 - 6*x ^8) + 27*x^6 - 9*x^7) + 81*x^6 - 27*x^7),x)
Output:
(3*log(log(3 - x) + 3)*(3*x^3 - x^4))/((x^3 + 2)*(exp(x)*(6*x^2 - 2*x^3 + 3*x^5 - x^6) - 9*x^4 + 3*x^5)) - (log(log(3 - x) + 3)*((9*x^4 - 3*x^5)/(2* x + x^4) - (exp(x)*(6*x^2 - 2*x^3 + 3*x^5 - x^6))/(2*x + x^4)))/(exp(x)*(6 *x^2 - 2*x^3 + 3*x^5 - x^6) - 9*x^4 + 3*x^5)
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-3 e^x x^3+e^{2 x} \left (2 x+x^4\right )+\left (e^{2 x} \left (18-6 x+36 x^3-12 x^4\right )+e^x \left (-81 x^2+54 x^3-9 x^4\right )+\left (e^{2 x} \left (6-2 x+12 x^3-4 x^4\right )+e^x \left (-27 x^2+18 x^3-3 x^4\right )\right ) \log (3-x)\right ) \log (3+\log (3-x))}{-81 x^6+27 x^7+e^x \left (108 x^4-36 x^5+54 x^7-18 x^8\right )+e^{2 x} \left (-36 x^2+12 x^3-36 x^5+12 x^6-9 x^8+3 x^9\right )+\left (-27 x^6+9 x^7+e^x \left (36 x^4-12 x^5+18 x^7-6 x^8\right )+e^{2 x} \left (-12 x^2+4 x^3-12 x^5+4 x^6-3 x^8+x^9\right )\right ) \log (3-x)} \, dx=\frac {e^{x} \mathrm {log}\left (\mathrm {log}\left (-x +3\right )+3\right )}{x \left (e^{x} x^{3}+2 e^{x}-3 x^{2}\right )} \] Input:
int(((((-4*x^4+12*x^3-2*x+6)*exp(x)^2+(-3*x^4+18*x^3-27*x^2)*exp(x))*log(3 -x)+(-12*x^4+36*x^3-6*x+18)*exp(x)^2+(-9*x^4+54*x^3-81*x^2)*exp(x))*log(lo g(3-x)+3)+(x^4+2*x)*exp(x)^2-3*exp(x)*x^3)/(((x^9-3*x^8+4*x^6-12*x^5+4*x^3 -12*x^2)*exp(x)^2+(-6*x^8+18*x^7-12*x^5+36*x^4)*exp(x)+9*x^7-27*x^6)*log(3 -x)+(3*x^9-9*x^8+12*x^6-36*x^5+12*x^3-36*x^2)*exp(x)^2+(-18*x^8+54*x^7-36* x^5+108*x^4)*exp(x)+27*x^7-81*x^6),x)
Output:
(e**x*log(log( - x + 3) + 3))/(x*(e**x*x**3 + 2*e**x - 3*x**2))