Integrand size = 121, antiderivative size = 27 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=2+e^{-2+\frac {4}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \] Output:
2+exp(4/ln((ln(x)-2*ln(2)+x^2)/(6+x))-2)
\[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx \] Input:
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]
Output:
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 |
| \(\displaystyle -4 \int \left (\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (x^3+12 x^2-\log (x) x+(1+\log (4)) x+6\right )}{6 x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}+\frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} \left (-x^3-12 x^2+\log (x) x-(1+\log (4)) x-6\right )}{6 (x+6) \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (24 \text {Subst}\left (\int \frac {e^{\frac {4}{\log \left (\frac {16 x^2+\log (x)}{4 x+6}\right )}-2}}{\left (16 x^2+\log (x)\right ) \log ^2\left (\frac {16 x^2+\log (x)}{4 x+6}\right )}dx,x,\frac {x}{4}\right )+4 (1+\log (4)) \text {Subst}\left (\int \frac {e^{\frac {4}{\log \left (\frac {16 x^2+\log (x)}{4 x+6}\right )}-2}}{(4 x+6) \left (16 x^2+\log (x)\right ) \log ^2\left (\frac {16 x^2+\log (x)}{4 x+6}\right )}dx,x,\frac {x}{4}\right )-148 \text {Subst}\left (\int \frac {e^{\frac {4}{\log \left (\frac {16 x^2+\log (x)}{4 x+6}\right )}-2}}{(4 x+6) \left (16 x^2+\log (x)\right ) \log ^2\left (\frac {16 x^2+\log (x)}{4 x+6}\right )}dx,x,\frac {x}{4}\right )-4 \text {Subst}\left (\int \frac {e^{\frac {4}{\log \left (\frac {16 x^2+\log (x)}{4 x+6}\right )}-2} \log (4 x)}{(4 x+6) \left (16 x^2+\log (x)\right ) \log ^2\left (\frac {16 x^2+\log (x)}{4 x+6}\right )}dx,x,\frac {x}{4}\right )+\int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2}}{x \left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx+\int \frac {e^{\frac {4}{\log \left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}-2} x}{\left (x^2+\log \left (\frac {x}{4}\right )\right ) \log ^2\left (\frac {x^2+\log \left (\frac {x}{4}\right )}{x+6}\right )}dx\right )\) |
Input:
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]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 433, normalized size of antiderivative = 16.04
\[{\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 )}}\]
Input:
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)
Output:
exp(-2*(I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)))^3+I*Pi*csgn(I/(6+x)* (-1/2*x^2+ln(2)-1/2*ln(x)))^2*csgn(I/(6+x))+I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln (2)-1/2*ln(x)))^2*csgn(I*(-1/2*x^2+ln(2)-1/2*ln(x)))-I*Pi*csgn(I/(6+x)*(-1 /2*x^2+ln(2)-1/2*ln(x)))*csgn(I/(6+x))*csgn(I*(-1/2*x^2+ln(2)-1/2*ln(x)))- 2*I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)))^2+2*I*Pi+2*ln(2)-2*ln(6+x) +2*ln(-1/2*x^2+ln(2)-1/2*ln(x))-4)/(I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2* ln(x)))^3+I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)))^2*csgn(I/(6+x))+I* Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)))^2*csgn(I*(-1/2*x^2+ln(2)-1/2*l n(x)))-I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)))*csgn(I/(6+x))*csgn(I* (-1/2*x^2+ln(2)-1/2*ln(x)))-2*I*Pi*csgn(I/(6+x)*(-1/2*x^2+ln(2)-1/2*ln(x)) )^2+2*I*Pi+2*ln(2)-2*ln(6+x)+2*ln(-1/2*x^2+ln(2)-1/2*ln(x))))
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=e^{\left (-\frac {2 \, {\left (\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right ) - 2\right )}}{\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )}\right )} \] Input:
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="fricas")
Output:
e^(-2*(log((x^2 - 2*log(2) + log(x))/(x + 6)) - 2)/log((x^2 - 2*log(2) + l og(x))/(x + 6)))
Time = 9.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=e^{\frac {4 - 2 \log {\left (\frac {x^{2} + \log {\left (x \right )} - 2 \log {\left (2 \right )}}{x + 6} \right )}}{\log {\left (\frac {x^{2} + \log {\left (x \right )} - 2 \log {\left (2 \right )}}{x + 6} \right )}}} \] Input:
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)
Output:
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 = 0.51 (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}} \] Input:
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="maxima")
Output:
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)
\[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\int { -\frac {4 \, {\left (x^{3} + 12 \, x^{2} + 2 \, x \log \left (2\right ) - x \log \left (x\right ) + x + 6\right )} e^{\left (-\frac {2 \, {\left (\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right ) - 2\right )}}{\log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )}\right )}}{{\left (x^{4} + 6 \, x^{3} - 2 \, {\left (x^{2} + 6 \, x\right )} \log \left (2\right ) + {\left (x^{2} + 6 \, x\right )} \log \left (x\right )\right )} \log \left (\frac {x^{2} - 2 \, \log \left (2\right ) + \log \left (x\right )}{x + 6}\right )^{2}} \,d x } \] Input:
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="giac")
Output:
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 = 3.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx={\mathrm {e}}^{-2}\,{\mathrm {e}}^{\frac {4}{\ln \left (\frac {\ln \left (\frac {x}{4}\right )+x^2}{x+6}\right )}} \] Input:
int(-(exp(-(2*log((log(x) - 2*log(2) + x^2)/(x + 6)) - 4)/log((log(x) - 2* log(2) + x^2)/(x + 6)))*(4*x + 8*x*log(2) - 4*x*log(x) + 48*x^2 + 4*x^3 + 24))/(log((log(x) - 2*log(2) + x^2)/(x + 6))^2*(log(x)*(6*x + x^2) + 6*x^3 + x^4 - 2*log(2)*(6*x + x^2))),x)
Output:
exp(-2)*exp(4/log((log(x/4) + x^2)/(x + 6)))
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4-2 \log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}{\log \left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )}} \left (-24-4 x-48 x^2-4 x^3-4 x \log (4)+4 x \log (x)\right )}{\left (6 x^3+x^4+\left (-6 x-x^2\right ) \log (4)+\left (6 x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {x^2-\log (4)+\log (x)}{6+x}\right )} \, dx=\frac {e^{\frac {4}{\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (2\right )+x^{2}}{x +6}\right )}}}{e^{2}} \] Input:
int((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)
Output:
e**(4/log((log(x) - 2*log(2) + x**2)/(x + 6)))/e**2