Integrand size = 61, antiderivative size = 24 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=24 e^{-(3-x)^2-x (-1+\log (\log (\log (x))))} x \] Output:
24/exp((3-x)^2+(ln(ln(ln(x)))-1)*x)*x
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=24 e^{-9+7 x-x^2} x \log ^{-x}(\log (x)) \] Input:
Integrate[(E^(-9 + 7*x - x^2 - x*Log[Log[Log[x]]])*(-24*x + (24 + 168*x - 48*x^2)*Log[x]*Log[Log[x]] - 24*x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]))/(L og[x]*Log[Log[x]]),x]
Output:
(24*E^(-9 + 7*x - x^2)*x)/Log[Log[x]]^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.
\(\displaystyle \int \frac {e^{-x^2+7 x-x \log (\log (\log (x)))-9} \left (\left (-48 x^2+168 x+24\right ) \log (x) \log (\log (x))-24 x-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {24 e^{-x^2+7 x-x \log (\log (\log (x)))-9} \left (2 x^2 \log (x) \log (\log (x))+x-7 x \log (x) \log (\log (x))-\log (x) \log (\log (x))\right )}{\log (x) \log (\log (x))}-24 x e^{-x^2+7 x-x \log (\log (\log (x)))-9} \log (\log (\log (x)))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 24 \int e^{-x^2-\log (\log (\log (x))) x+7 x-9}dx+168 \int e^{-x^2-\log (\log (\log (x))) x+7 x-9} xdx-48 \int e^{-x^2-\log (\log (\log (x))) x+7 x-9} x^2dx-24 \int \frac {e^{-x^2-\log (\log (\log (x))) x+7 x-9} x}{\log (x) \log (\log (x))}dx-24 \int e^{-x^2-\log (\log (\log (x))) x+7 x-9} x \log (\log (\log (x)))dx\) |
Input:
Int[(E^(-9 + 7*x - x^2 - x*Log[Log[Log[x]]])*(-24*x + (24 + 168*x - 48*x^2 )*Log[x]*Log[Log[x]] - 24*x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]))/(Log[x]* Log[Log[x]]),x]
Output:
$Aborted
Time = 0.87 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(24 x \,{\mathrm e}^{-x \ln \left (\ln \left (\ln \left (x \right )\right )\right )-x^{2}+7 x -9}\) | \(21\) |
risch | \(24 x \ln \left (\ln \left (x \right )\right )^{-x} {\mathrm e}^{-x^{2}+7 x -9}\) | \(22\) |
Input:
int((-24*x*ln(x)*ln(ln(x))*ln(ln(ln(x)))+(-48*x^2+168*x+24)*ln(x)*ln(ln(x) )-24*x)/ln(x)/ln(ln(x))/exp(x*ln(ln(ln(x)))+x^2-7*x+9),x,method=_RETURNVER BOSE)
Output:
24*x/exp(x*ln(ln(ln(x)))+x^2-7*x+9)
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=24 \, x e^{\left (-x^{2} - x \log \left (\log \left (\log \left (x\right )\right )\right ) + 7 \, x - 9\right )} \] Input:
integrate((-24*x*log(x)*log(log(x))*log(log(log(x)))+(-48*x^2+168*x+24)*lo g(x)*log(log(x))-24*x)/log(x)/log(log(x))/exp(x*log(log(log(x)))+x^2-7*x+9 ),x, algorithm="fricas")
Output:
24*x*e^(-x^2 - x*log(log(log(x))) + 7*x - 9)
Time = 8.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=24 x e^{- x^{2} - x \log {\left (\log {\left (\log {\left (x \right )} \right )} \right )} + 7 x - 9} \] Input:
integrate((-24*x*ln(x)*ln(ln(x))*ln(ln(ln(x)))+(-48*x**2+168*x+24)*ln(x)*l n(ln(x))-24*x)/ln(x)/ln(ln(x))/exp(x*ln(ln(ln(x)))+x**2-7*x+9),x)
Output:
24*x*exp(-x**2 - x*log(log(log(x))) + 7*x - 9)
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=24 \, x e^{\left (-x^{2} - x \log \left (\log \left (\log \left (x\right )\right )\right ) + 7 \, x - 9\right )} \] Input:
integrate((-24*x*log(x)*log(log(x))*log(log(log(x)))+(-48*x^2+168*x+24)*lo g(x)*log(log(x))-24*x)/log(x)/log(log(x))/exp(x*log(log(log(x)))+x^2-7*x+9 ),x, algorithm="maxima")
Output:
24*x*e^(-x^2 - x*log(log(log(x))) + 7*x - 9)
\[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=\int { -\frac {24 \, {\left (x \log \left (x\right ) \log \left (\log \left (x\right )\right ) \log \left (\log \left (\log \left (x\right )\right )\right ) + {\left (2 \, x^{2} - 7 \, x - 1\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + x\right )} e^{\left (-x^{2} - x \log \left (\log \left (\log \left (x\right )\right )\right ) + 7 \, x - 9\right )}}{\log \left (x\right ) \log \left (\log \left (x\right )\right )} \,d x } \] Input:
integrate((-24*x*log(x)*log(log(x))*log(log(log(x)))+(-48*x^2+168*x+24)*lo g(x)*log(log(x))-24*x)/log(x)/log(log(x))/exp(x*log(log(log(x)))+x^2-7*x+9 ),x, algorithm="giac")
Output:
undef
Time = 2.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=\frac {24\,x\,{\mathrm {e}}^{7\,x}\,{\mathrm {e}}^{-9}\,{\mathrm {e}}^{-x^2}}{{\ln \left (\ln \left (x\right )\right )}^x} \] Input:
int(-(exp(7*x - x*log(log(log(x))) - x^2 - 9)*(24*x - log(log(x))*log(x)*( 168*x - 48*x^2 + 24) + 24*x*log(log(x))*log(log(log(x)))*log(x)))/(log(log (x))*log(x)),x)
Output:
(24*x*exp(7*x)*exp(-9)*exp(-x^2))/log(log(x))^x
\[ \int \frac {e^{-9+7 x-x^2-x \log (\log (\log (x)))} \left (-24 x+\left (24+168 x-48 x^2\right ) \log (x) \log (\log (x))-24 x \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx=\frac {24 \left (\int \frac {e^{7 x}}{e^{x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{x}}d x \right )-48 \left (\int \frac {e^{7 x} x^{2}}{e^{x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{x}}d x \right )-24 \left (\int \frac {e^{7 x} \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )\right ) x}{e^{x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{x}}d x \right )-24 \left (\int \frac {e^{7 x} x}{e^{x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right )}d x \right )+168 \left (\int \frac {e^{7 x} x}{e^{x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{x}}d x \right )}{e^{9}} \] Input:
int((-24*x*log(x)*log(log(x))*log(log(log(x)))+(-48*x^2+168*x+24)*log(x)*l og(log(x))-24*x)/log(x)/log(log(x))/exp(x*log(log(log(x)))+x^2-7*x+9),x)
Output:
(24*(int(e**(7*x)/(e**(x**2)*log(log(x))**x),x) - 2*int((e**(7*x)*x**2)/(e **(x**2)*log(log(x))**x),x) - int((e**(7*x)*log(log(log(x)))*x)/(e**(x**2) *log(log(x))**x),x) - int((e**(7*x)*x)/(e**(x**2)*log(log(x))**x*log(log(x ))*log(x)),x) + 7*int((e**(7*x)*x)/(e**(x**2)*log(log(x))**x),x)))/e**9