Integrand size = 109, antiderivative size = 18 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=8 e^{x+\frac {1}{e^x-4 \log (6 x)}} \end {dmath*}
Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=8 e^{x+\frac {1}{e^x-4 \log (6 x)}} \end {dmath*}
Integrate[(E^((-1 + E^x*(-x - Log[2]) + (4*x + 4*Log[2])*Log[6*x])/(-E^x + 4*Log[6*x]))*(16 - 4*E^x*x + 4*E^(2*x)*x - 32*E^x*x*Log[6*x] + 64*x*Log[6 *x]^2))/(E^(2*x)*x - 8*E^x*x*Log[6*x] + 16*x*Log[6*x]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-4 e^x x+4 e^{2 x} x+64 x \log ^2(6 x)-32 e^x x \log (6 x)+16\right ) \exp \left (\frac {e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)-1}{4 \log (6 x)-e^x}\right )}{e^{2 x} x+16 x \log ^2(6 x)-8 e^x x \log (6 x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 \left (-e^x x+e^{2 x} x+16 x \log ^2(6 x)-8 e^x x \log (6 x)+4\right ) \exp \left (\frac {e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)-1}{4 \log (6 x)-e^x}\right )}{x \left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}-1} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2^{2-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}} 3^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}-1}}{\left (e^x-4 \log (6 x)\right )^2}+6^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}}-\frac {6^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{-\frac {4 x+4 \log (2)}{e^x-4 \log (6 x)}}}{e^x-4 \log (6 x)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}-\frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}+1}}{e^x-4 \log (6 x)}-\frac {2^{2-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} 3^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} (x \log (6 x)-1) x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}}}{\left (e^x-4 \log (6 x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {6^{-\frac {4 (x+\log (2))}{e^x-4 \log (6 x)}} e^{\frac {e^x (x+\log (2))+1}{e^x-4 \log (6 x)}} x^{\frac {-4 x-e^x+4 \log (x)+\log (81)}{e^x-4 \log (6 x)}} \left (16 x \log ^2(6 x)-8 e^x x \log (6 x)-e^x x+e^{2 x} x+4\right )}{\left (e^x-4 \log (6 x)\right )^2}dx\) |
Int[(E^((-1 + E^x*(-x - Log[2]) + (4*x + 4*Log[2])*Log[6*x])/(-E^x + 4*Log [6*x]))*(16 - 4*E^x*x + 4*E^(2*x)*x - 32*E^x*x*Log[6*x] + 64*x*Log[6*x]^2) )/(E^(2*x)*x - 8*E^x*x*Log[6*x] + 16*x*Log[6*x]^2),x]
3.11.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).
Time = 1.70 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33
| method | result | size |
| risch | \(4 \,{\mathrm e}^{\frac {{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} x -4 \ln \left (6 x \right ) \ln \left (2\right )-4 x \ln \left (6 x \right )+1}{{\mathrm e}^{x}-4 \ln \left (6 x \right )}}\) | \(42\) |
| parallelrisch | \(4 \,{\mathrm e}^{\frac {-\left (4 \ln \left (2\right )+4 x \right ) \ln \left (6 x \right )-\left (-x -\ln \left (2\right )\right ) {\mathrm e}^{x}+1}{{\mathrm e}^{x}-4 \ln \left (6 x \right )}}\) | \(44\) |
int((64*x*ln(6*x)^2-32*x*exp(x)*ln(6*x)+4*x*exp(x)^2-4*exp(x)*x+16)*exp((( 4*ln(2)+4*x)*ln(6*x)+(-x-ln(2))*exp(x)-1)/(4*ln(6*x)-exp(x)))/(16*x*ln(6*x )^2-8*x*exp(x)*ln(6*x)+x*exp(x)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=4 \, e^{\left (\frac {{\left (x + \log \left (2\right )\right )} e^{x} - 4 \, {\left (x + \log \left (2\right )\right )} \log \left (6 \, x\right ) + 1}{e^{x} - 4 \, \log \left (6 \, x\right )}\right )} \end {dmath*}
integrate((64*x*log(6*x)^2-32*x*exp(x)*log(6*x)+4*x*exp(x)^2-4*exp(x)*x+16 )*exp(((4*log(2)+4*x)*log(6*x)+(-x-log(2))*exp(x)-1)/(4*log(6*x)-exp(x)))/ (16*x*log(6*x)^2-8*x*exp(x)*log(6*x)+x*exp(x)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=4 e^{\frac {\left (- x - \log {\left (2 \right )}\right ) e^{x} + \left (4 x + 4 \log {\left (2 \right )}\right ) \log {\left (6 x \right )} - 1}{- e^{x} + 4 \log {\left (6 x \right )}}} \end {dmath*}
integrate((64*x*ln(6*x)**2-32*x*exp(x)*ln(6*x)+4*x*exp(x)**2-4*exp(x)*x+16 )*exp(((4*ln(2)+4*x)*ln(6*x)+(-x-ln(2))*exp(x)-1)/(4*ln(6*x)-exp(x)))/(16* x*ln(6*x)**2-8*x*exp(x)*ln(6*x)+x*exp(x)**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (18) = 36\).
Time = 0.63 (sec) , antiderivative size = 199, normalized size of antiderivative = 11.06 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=4 \, e^{\left (\frac {x e^{x}}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, x \log \left (3\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, x \log \left (2\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} + \frac {e^{x} \log \left (2\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, \log \left (3\right ) \log \left (2\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, \log \left (2\right )^{2}}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, x \log \left (x\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} - \frac {4 \, \log \left (2\right ) \log \left (x\right )}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )} + \frac {1}{e^{x} - 4 \, \log \left (3\right ) - 4 \, \log \left (2\right ) - 4 \, \log \left (x\right )}\right )} \end {dmath*}
integrate((64*x*log(6*x)^2-32*x*exp(x)*log(6*x)+4*x*exp(x)^2-4*exp(x)*x+16 )*exp(((4*log(2)+4*x)*log(6*x)+(-x-log(2))*exp(x)-1)/(4*log(6*x)-exp(x)))/ (16*x*log(6*x)^2-8*x*exp(x)*log(6*x)+x*exp(x)^2),x, algorithm=\
4*e^(x*e^x/(e^x - 4*log(3) - 4*log(2) - 4*log(x)) - 4*x*log(3)/(e^x - 4*lo g(3) - 4*log(2) - 4*log(x)) - 4*x*log(2)/(e^x - 4*log(3) - 4*log(2) - 4*lo g(x)) + e^x*log(2)/(e^x - 4*log(3) - 4*log(2) - 4*log(x)) - 4*log(3)*log(2 )/(e^x - 4*log(3) - 4*log(2) - 4*log(x)) - 4*log(2)^2/(e^x - 4*log(3) - 4* log(2) - 4*log(x)) - 4*x*log(x)/(e^x - 4*log(3) - 4*log(2) - 4*log(x)) - 4 *log(2)*log(x)/(e^x - 4*log(3) - 4*log(2) - 4*log(x)) + 1/(e^x - 4*log(3) - 4*log(2) - 4*log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
Time = 1.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=4 \, e^{\left (\frac {x e^{x} - 4 \, x \log \left (2\right ) + e^{x} \log \left (2\right ) - 4 \, \log \left (2\right )^{2} - 4 \, x \log \left (3 \, x\right ) - 4 \, \log \left (2\right ) \log \left (3 \, x\right ) + 1}{e^{x} - 4 \, \log \left (2\right ) - 4 \, \log \left (3 \, x\right )}\right )} \end {dmath*}
integrate((64*x*log(6*x)^2-32*x*exp(x)*log(6*x)+4*x*exp(x)^2-4*exp(x)*x+16 )*exp(((4*log(2)+4*x)*log(6*x)+(-x-log(2))*exp(x)-1)/(4*log(6*x)-exp(x)))/ (16*x*log(6*x)^2-8*x*exp(x)*log(6*x)+x*exp(x)^2),x, algorithm=\
4*e^((x*e^x - 4*x*log(2) + e^x*log(2) - 4*log(2)^2 - 4*x*log(3*x) - 4*log( 2)*log(3*x) + 1)/(e^x - 4*log(2) - 4*log(3*x)))
Timed out. \begin {dmath*} \int \frac {e^{\frac {-1+e^x (-x-\log (2))+(4 x+4 \log (2)) \log (6 x)}{-e^x+4 \log (6 x)}} \left (16-4 e^x x+4 e^{2 x} x-32 e^x x \log (6 x)+64 x \log ^2(6 x)\right )}{e^{2 x} x-8 e^x x \log (6 x)+16 x \log ^2(6 x)} \, dx=\int \frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^x\,\left (x+\ln \left (2\right )\right )-\ln \left (6\,x\right )\,\left (4\,x+4\,\ln \left (2\right )\right )+1}{4\,\ln \left (6\,x\right )-{\mathrm {e}}^x}}\,\left (64\,x\,{\ln \left (6\,x\right )}^2-32\,x\,{\mathrm {e}}^x\,\ln \left (6\,x\right )+4\,x\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^x+16\right )}{16\,x\,{\ln \left (6\,x\right )}^2-8\,x\,{\mathrm {e}}^x\,\ln \left (6\,x\right )+x\,{\mathrm {e}}^{2\,x}} \,d x \end {dmath*}
int((exp(-(exp(x)*(x + log(2)) - log(6*x)*(4*x + 4*log(2)) + 1)/(4*log(6*x ) - exp(x)))*(4*x*exp(2*x) + 64*x*log(6*x)^2 - 4*x*exp(x) - 32*x*log(6*x)* exp(x) + 16))/(x*exp(2*x) + 16*x*log(6*x)^2 - 8*x*log(6*x)*exp(x)),x)