Integrand size = 121, antiderivative size = 27 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=2+e^{-2+\frac {4}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \]
\[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx \]
Integrate[(E^((4 - 2*Log[(x^2 - Log[4] + Log[x])/(6 + x)])/Log[(x^2 - Log[ 4] + Log[x])/(6 + x)])*(-24 - 4*x - 48*x^2 - 4*x^3 - 4*x*Log[4] + 4*x*Log[ x]))/((6*x^3 + x^4 + (-6*x - x^2)*Log[4] + (6*x + x^2)*Log[x])*Log[(x^2 - Log[4] + Log[x])/(6 + x)]^2),x]
Integrate[(E^((4 - 2*Log[(x^2 - Log[4] + Log[x])/(6 + x)])/Log[(x^2 - Log[ 4] + Log[x])/(6 + x)])*(-24 - 4*x - 48*x^2 - 4*x^3 - 4*x*Log[4] + 4*x*Log[ x]))/((6*x^3 + x^4 + (-6*x - x^2)*Log[4] + (6*x + x^2)*Log[x])*Log[(x^2 - Log[4] + Log[x])/(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.
| \(\displaystyle \int \frac {\left (-4 x^3-48 x^2-4 x+4 x \log (x)-4 x \log (4)-24\right ) \exp \left (\frac {4-2 \log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}\right )}{\left (x^4+6 x^3+\left (x^2+6 x\right ) \log (x)+\left (-x^2-6 x\right ) \log (4)\right ) \log ^2\left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (-4 x^3-48 x^2+4 x \log (x)+x (-4-4 \log (4))-24\right ) \exp \left (\frac {4-2 \log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}\right )}{\left (x^4+6 x^3+\left (x^2+6 x\right ) \log (x)+\left (-x^2-6 x\right ) \log (4)\right ) \log ^2\left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 \left (-x^3-12 x^2+x \log (x)-x (1+\log (4))-6\right ) \exp \left (-\frac {2 \left (\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )-2\right )}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {e^{\frac {4}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}} \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )^{-\frac {2}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}} \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )^{-\frac {2}{\log \left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log (x)-\log (4)}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{x (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
Int[(E^((4 - 2*Log[(x^2 - Log[4] + Log[x])/(6 + x)])/Log[(x^2 - Log[4] + L og[x])/(6 + x)])*(-24 - 4*x - 48*x^2 - 4*x^3 - 4*x*Log[4] + 4*x*Log[x]))/( (6*x^3 + x^4 + (-6*x - x^2)*Log[4] + (6*x + x^2)*Log[x])*Log[(x^2 - Log[4] + Log[x])/(6 + x)]^2),x]
3.24.40.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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]]
Time = 266.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
| method | result | size |
| parallelrisch | \({\mathrm e}^{-\frac {2 \left (\ln \left (\frac {\ln \left (x \right )-2 \ln \left (2\right )+x^{2}}{6+x}\right )-2\right )}{\ln \left (\frac {\ln \left (x \right )-2 \ln \left (2\right )+x^{2}}{6+x}\right )}}\) | \(42\) |
| risch | \({\mathrm e}^{-\frac {2 \left (i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{3}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{6+x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2} \operatorname {csgn}\left (i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right ) \operatorname {csgn}\left (\frac {i}{6+x}\right ) \operatorname {csgn}\left (i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )\right )-2 i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2}+2 i \pi +2 \ln \left (2\right )-2 \ln \left (6+x \right )+2 \ln \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )-4\right )}{i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{3}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{6+x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2} \operatorname {csgn}\left (i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right ) \operatorname {csgn}\left (\frac {i}{6+x}\right ) \operatorname {csgn}\left (i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )\right )-2 i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}{6+x}\right )}^{2}+2 i \pi +2 \ln \left (2\right )-2 \ln \left (6+x \right )+2 \ln \left (-\frac {x^{2}}{2}+\ln \left (2\right )-\frac {\ln \left (x \right )}{2}\right )}}\) | \(433\) |
int((4*x*ln(x)-8*x*ln(2)-4*x^3-48*x^2-4*x-24)*exp((-2*ln((ln(x)-2*ln(2)+x^ 2)/(6+x))+4)/ln((ln(x)-2*ln(2)+x^2)/(6+x)))/((x^2+6*x)*ln(x)+2*(-x^2-6*x)* ln(2)+x^4+6*x^3)/ln((ln(x)-2*ln(2)+x^2)/(6+x))^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=e^{\left (-\frac {2 \, {\left (\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right ) - 2\right )}}{\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )}\right )} \]
integrate((4*x*log(x)-8*x*log(2)-4*x^3-48*x^2-4*x-24)*exp((-2*log((log(x)- 2*log(2)+x^2)/(6+x))+4)/log((log(x)-2*log(2)+x^2)/(6+x)))/((x^2+6*x)*log(x )+2*(-x^2-6*x)*log(2)+x^4+6*x^3)/log((log(x)-2*log(2)+x^2)/(6+x))^2,x, alg orithm=\
Time = 8.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=e^{\frac {4 - 2 \log {\left (\frac {x^{2} + \log {\left (x \right )} - 2 \log {\left (2 \right )}}{x + 6} \right )}}{\log {\left (\frac {x^{2} + \log {\left (x \right )} - 2 \log {\left (2 \right )}}{x + 6} \right )}}} \]
integrate((4*x*ln(x)-8*x*ln(2)-4*x**3-48*x**2-4*x-24)*exp((-2*ln((ln(x)-2* ln(2)+x**2)/(6+x))+4)/ln((ln(x)-2*ln(2)+x**2)/(6+x)))/((x**2+6*x)*ln(x)+2* (-x**2-6*x)*ln(2)+x**4+6*x**3)/ln((ln(x)-2*ln(2)+x**2)/(6+x))**2,x)
exp((4 - 2*log((x**2 + log(x) - 2*log(2))/(x + 6)))/log((x**2 + log(x) - 2 *log(2))/(x + 6)))
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (26) = 52\).
Time = 0.60 (sec) , antiderivative size = 384, normalized size of antiderivative = 14.22 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\frac {x^{3} e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )}}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} + \frac {12 \, x^{2} e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )}}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} + \frac {2 \, x e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )} \log \left (2\right )}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} - \frac {x e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )} \log \left (x\right )}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} + \frac {x e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )}}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} + \frac {6 \, e^{\left (\frac {4}{\log \left (x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x + 6\right )}\right )}}{x^{3} e^{2} + 12 \, x^{2} e^{2} + x {\left (2 \, \log \left (2\right ) + 1\right )} e^{2} - x e^{2} \log \left (x\right ) + 6 \, e^{2}} \]
integrate((4*x*log(x)-8*x*log(2)-4*x^3-48*x^2-4*x-24)*exp((-2*log((log(x)- 2*log(2)+x^2)/(6+x))+4)/log((log(x)-2*log(2)+x^2)/(6+x)))/((x^2+6*x)*log(x )+2*(-x^2-6*x)*log(2)+x^4+6*x^3)/log((log(x)-2*log(2)+x^2)/(6+x))^2,x, alg orithm=\
x^3*e^(4/(log(x^2 - 2*log(2) + log(x)) - log(x + 6)))/(x^3*e^2 + 12*x^2*e^ 2 + x*(2*log(2) + 1)*e^2 - x*e^2*log(x) + 6*e^2) + 12*x^2*e^(4/(log(x^2 - 2*log(2) + log(x)) - log(x + 6)))/(x^3*e^2 + 12*x^2*e^2 + x*(2*log(2) + 1) *e^2 - x*e^2*log(x) + 6*e^2) + 2*x*e^(4/(log(x^2 - 2*log(2) + log(x)) - lo g(x + 6)))*log(2)/(x^3*e^2 + 12*x^2*e^2 + x*(2*log(2) + 1)*e^2 - x*e^2*log (x) + 6*e^2) - x*e^(4/(log(x^2 - 2*log(2) + log(x)) - log(x + 6)))*log(x)/ (x^3*e^2 + 12*x^2*e^2 + x*(2*log(2) + 1)*e^2 - x*e^2*log(x) + 6*e^2) + x*e ^(4/(log(x^2 - 2*log(2) + log(x)) - log(x + 6)))/(x^3*e^2 + 12*x^2*e^2 + x *(2*log(2) + 1)*e^2 - x*e^2*log(x) + 6*e^2) + 6*e^(4/(log(x^2 - 2*log(2) + log(x)) - log(x + 6)))/(x^3*e^2 + 12*x^2*e^2 + x*(2*log(2) + 1)*e^2 - x*e ^2*log(x) + 6*e^2)
\[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\int { -\frac {4 \, {\left (x^{3} + 12 \, x^{2} + 2 \, x \log \left (2\right ) - x \log \left (x\right ) + x + 6\right )} e^{\left (-\frac {2 \, {\left (\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right ) - 2\right )}}{\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )}\right )}}{{\left (x^{4} + 6 \, x^{3} - 2 \, {\left (x^{2} + 6 \, x\right )} \log \left (2\right ) + {\left (x^{2} + 6 \, x\right )} \log \left (x\right )\right )} \log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )^{2}} \,d x } \]
integrate((4*x*log(x)-8*x*log(2)-4*x^3-48*x^2-4*x-24)*exp((-2*log((log(x)- 2*log(2)+x^2)/(6+x))+4)/log((log(x)-2*log(2)+x^2)/(6+x)))/((x^2+6*x)*log(x )+2*(-x^2-6*x)*log(2)+x^4+6*x^3)/log((log(x)-2*log(2)+x^2)/(6+x))^2,x, alg orithm=\
integrate(-4*(x^3 + 12*x^2 + 2*x*log(2) - x*log(x) + x + 6)*e^(-2*(log((x^ 2 - 2*log(2) + log(x))/(x + 6)) - 2)/log((x^2 - 2*log(2) + log(x))/(x + 6) ))/((x^4 + 6*x^3 - 2*(x^2 + 6*x)*log(2) + (x^2 + 6*x)*log(x))*log((x^2 - 2 *log(2) + log(x))/(x + 6))^2), x)
Time = 11.89 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx={\mathrm {e}}^{-2}\,{\mathrm {e}}^{\frac {4}{\ln \left (\frac {\ln \left (\frac {x}{4}\right )+x^2}{x+6}\right )}} \]