Integrand size = 93, antiderivative size = 24 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=e^{(4-x) \left (-2+2 x+\left (-3+\log \left (\frac {\log (x)}{5}\right )\right )^2\right )} \] Output:
exp((4-x)*(2*x-2+(ln(1/5*ln(x))-3)^2))
\[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=\int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx \] Input:
Integrate[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log [x]/5]^2)*(-24 + 6*x + (x - 4*x^2)*Log[x] + (8 - 2*x + 6*x*Log[x])*Log[Log [x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*Log[x]),x]
Output:
Integrate[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log [x]/5]^2)*(-24 + 6*x + (x - 4*x^2)*Log[x] + (8 - 2*x + 6*x*Log[x])*Log[Log [x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*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.
| \(\displaystyle \int \frac {\left (\left (x-4 x^2\right ) \log (x)+6 x-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )+(-2 x+6 x \log (x)+8) \log \left (\frac {\log (x)}{5}\right )-24\right ) \exp \left (-2 x^2+x+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )+(6 x-24) \log \left (\frac {\log (x)}{5}\right )+28\right )}{x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\left (x-4 x^2\right ) \log (x)+6 x-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )+(-2 x+6 x \log (x)+8) \log \left (\frac {\log (x)}{5}\right )-24\right ) \exp \left (-\left ((x-4) \left (2 x+\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+7\right )\right )\right )}{x \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (-4 x^2 \log (x)+6 x+x \log (x)-24\right ) \exp \left (-\left ((x-4) \left (2 x+\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+7\right )\right )\right )}{x \log (x)}+\log ^2\left (\frac {\log (x)}{5}\right ) \left (-\exp \left (-\left ((x-4) \left (2 x+\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+7\right )\right )\right )\right )+\frac {2 (-x+3 x \log (x)+4) \log \left (\frac {\log (x)}{5}\right ) \exp \left (-\left ((x-4) \left (2 x+\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+7\right )\right )\right )}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right )dx-4 \int \exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right ) xdx+6 \int \frac {\exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right )}{\log (x)}dx-24 \int \frac {\exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right )}{x \log (x)}dx+6 \int \exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right ) \log \left (\frac {\log (x)}{5}\right )dx-2 \int \frac {\exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{\log (x)}dx+8 \int \frac {\exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right ) \log \left (\frac {\log (x)}{5}\right )}{x \log (x)}dx-\int \exp \left (-\left ((x-4) \left (\log ^2\left (\frac {\log (x)}{5}\right )-6 \log \left (\frac {\log (x)}{5}\right )+2 x+7\right )\right )\right ) \log ^2\left (\frac {\log (x)}{5}\right )dx\) |
Input:
Int[(E^(28 + x - 2*x^2 + (-24 + 6*x)*Log[Log[x]/5] + (4 - x)*Log[Log[x]/5] ^2)*(-24 + 6*x + (x - 4*x^2)*Log[x] + (8 - 2*x + 6*x*Log[x])*Log[Log[x]/5] - x*Log[x]*Log[Log[x]/5]^2))/(x*Log[x]),x]
Output:
$Aborted
Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(\left (\frac {\ln \left (x \right )}{5}\right )^{6 x -24} {\mathrm e}^{-\left (x -4\right ) \left (\ln \left (\frac {\ln \left (x \right )}{5}\right )^{2}+2 x +7\right )}\) | \(30\) |
| parallelrisch | \({\mathrm e}^{\left (-x +4\right ) \ln \left (\frac {\ln \left (x \right )}{5}\right )^{2}+\left (6 x -24\right ) \ln \left (\frac {\ln \left (x \right )}{5}\right )-2 x^{2}+x +28}\) | \(34\) |
Input:
int((-x*ln(x)*ln(1/5*ln(x))^2+(6*x*ln(x)-2*x+8)*ln(1/5*ln(x))+(-4*x^2+x)*l n(x)+6*x-24)*exp((-x+4)*ln(1/5*ln(x))^2+(6*x-24)*ln(1/5*ln(x))-2*x^2+x+28) /x/ln(x),x,method=_RETURNVERBOSE)
Output:
(1/5*ln(x))^(6*x-24)*exp(-(x-4)*(ln(1/5*ln(x))^2+2*x+7))
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=e^{\left (-{\left (x - 4\right )} \log \left (\frac {1}{5} \, \log \left (x\right )\right )^{2} - 2 \, x^{2} + 6 \, {\left (x - 4\right )} \log \left (\frac {1}{5} \, \log \left (x\right )\right ) + x + 28\right )} \] Input:
integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+ (-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)*log(1/5*log(x))^2+(6*x-24)*log(1/5*lo g(x))-2*x^2+x+28)/x/log(x),x, algorithm="fricas")
Output:
e^(-(x - 4)*log(1/5*log(x))^2 - 2*x^2 + 6*(x - 4)*log(1/5*log(x)) + x + 28 )
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=e^{- 2 x^{2} + x + \left (4 - x\right ) \log {\left (\frac {\log {\left (x \right )}}{5} \right )}^{2} + \left (6 x - 24\right ) \log {\left (\frac {\log {\left (x \right )}}{5} \right )} + 28} \] Input:
integrate((-x*ln(x)*ln(1/5*ln(x))**2+(6*x*ln(x)-2*x+8)*ln(1/5*ln(x))+(-4*x **2+x)*ln(x)+6*x-24)*exp((-x+4)*ln(1/5*ln(x))**2+(6*x-24)*ln(1/5*ln(x))-2* x**2+x+28)/x/ln(x),x)
Output:
exp(-2*x**2 + x + (4 - x)*log(log(x)/5)**2 + (6*x - 24)*log(log(x)/5) + 28 )
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.88 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=\frac {59604644775390625 \, e^{\left (-x \log \left (5\right )^{2} + 2 \, x \log \left (5\right ) \log \left (\log \left (x\right )\right ) - x \log \left (\log \left (x\right )\right )^{2} - 2 \, x^{2} - 6 \, x \log \left (5\right ) + 4 \, \log \left (5\right )^{2} + 6 \, x \log \left (\log \left (x\right )\right ) - 8 \, \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 4 \, \log \left (\log \left (x\right )\right )^{2} + x + 28\right )}}{\log \left (x\right )^{24}} \] Input:
integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+ (-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)*log(1/5*log(x))^2+(6*x-24)*log(1/5*lo g(x))-2*x^2+x+28)/x/log(x),x, algorithm="maxima")
Output:
59604644775390625*e^(-x*log(5)^2 + 2*x*log(5)*log(log(x)) - x*log(log(x))^ 2 - 2*x^2 - 6*x*log(5) + 4*log(5)^2 + 6*x*log(log(x)) - 8*log(5)*log(log(x )) + 4*log(log(x))^2 + x + 28)/log(x)^24
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=e^{\left (-x \log \left (\frac {1}{5} \, \log \left (x\right )\right )^{2} - 2 \, x^{2} + 6 \, x \log \left (\frac {1}{5} \, \log \left (x\right )\right ) + 4 \, \log \left (\frac {1}{5} \, \log \left (x\right )\right )^{2} + x - 24 \, \log \left (\frac {1}{5} \, \log \left (x\right )\right ) + 28\right )} \] Input:
integrate((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+ (-4*x^2+x)*log(x)+6*x-24)*exp((-x+4)*log(1/5*log(x))^2+(6*x-24)*log(1/5*lo g(x))-2*x^2+x+28)/x/log(x),x, algorithm="giac")
Output:
e^(-x*log(1/5*log(x))^2 - 2*x^2 + 6*x*log(1/5*log(x)) + 4*log(1/5*log(x))^ 2 + x - 24*log(1/5*log(x)) + 28)
Time = 3.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.25 \[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=\frac {59604644775390625\,{\mathrm {e}}^{-x\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{28}\,{\mathrm {e}}^{4\,{\ln \left (\ln \left (x\right )\right )}^2}\,{\mathrm {e}}^{4\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-x\,{\ln \left (\ln \left (x\right )\right )}^2}\,{\ln \left (x\right )}^{6\,x}\,{\ln \left (x\right )}^{2\,x\,\ln \left (5\right )}}{5^{6\,x}\,{\ln \left (x\right )}^{8\,\ln \left (5\right )}\,{\ln \left (x\right )}^{24}} \] Input:
int((exp(x + log(log(x)/5)*(6*x - 24) - log(log(x)/5)^2*(x - 4) - 2*x^2 + 28)*(6*x + log(log(x)/5)*(6*x*log(x) - 2*x + 8) + log(x)*(x - 4*x^2) - x*l og(log(x)/5)^2*log(x) - 24))/(x*log(x)),x)
Output:
(59604644775390625*exp(-x*log(5)^2)*exp(28)*exp(4*log(log(x))^2)*exp(4*log (5)^2)*exp(-2*x^2)*exp(x)*exp(-x*log(log(x))^2)*log(x)^(6*x)*log(x)^(2*x*l og(5)))/(5^(6*x)*log(x)^(8*log(5))*log(x)^24)
\[ \int \frac {e^{28+x-2 x^2+(-24+6 x) \log \left (\frac {\log (x)}{5}\right )+(4-x) \log ^2\left (\frac {\log (x)}{5}\right )} \left (-24+6 x+\left (x-4 x^2\right ) \log (x)+(8-2 x+6 x \log (x)) \log \left (\frac {\log (x)}{5}\right )-x \log (x) \log ^2\left (\frac {\log (x)}{5}\right )\right )}{x \log (x)} \, dx=\int \frac {\left (-x \,\mathrm {log}\left (x \right ) \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{5}\right )^{2}+\left (6 \,\mathrm {log}\left (x \right ) x -2 x +8\right ) \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{5}\right )+\left (-4 x^{2}+x \right ) \mathrm {log}\left (x \right )+6 x -24\right ) {\mathrm e}^{\left (-x +4\right ) \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{5}\right )^{2}+\left (6 x -24\right ) \mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{5}\right )-2 x^{2}+x +28}}{x \,\mathrm {log}\left (x \right )}d x \] Input:
int((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+(-4*x^ 2+x)*log(x)+6*x-24)*exp((-x+4)*log(1/5*log(x))^2+(6*x-24)*log(1/5*log(x))- 2*x^2+x+28)/x/log(x),x)
Output:
int((-x*log(x)*log(1/5*log(x))^2+(6*x*log(x)-2*x+8)*log(1/5*log(x))+(-4*x^ 2+x)*log(x)+6*x-24)*exp((-x+4)*log(1/5*log(x))^2+(6*x-24)*log(1/5*log(x))- 2*x^2+x+28)/x/log(x),x)