Integrand size = 83, antiderivative size = 23 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \] Output:
exp(1/10000*x^5/ln(1/5*x)^4*ln(x)^4-3)^4
Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \] Input:
Integrate[(E^((-30000*Log[x/5]^4 + x^5*Log[x]^4)/(2500*Log[x/5]^4))*(4*x^4 *Log[x/5]*Log[x]^3 + (-4*x^4 + 5*x^4*Log[x/5])*Log[x]^4))/(2500*Log[x/5]^5 ),x]
Output:
E^(-12 + (x^5*Log[x]^4)/(2500*Log[x/5]^4))
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.
| \(\displaystyle \int \frac {e^{\frac {x^5 \log ^4(x)-30000 \log ^4\left (\frac {x}{5}\right )}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (\left (5 x^4 \log \left (\frac {x}{5}\right )-4 x^4\right ) \log ^4(x)+4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)-\left (4 x^4-5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{\log ^5\left (\frac {x}{5}\right )}dx}{2500}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\int \frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \log ^3(x) \left (5 \log (x) \log \left (\frac {x}{5}\right )+4 \log \left (\frac {x}{5}\right )-4 \log (x)\right )}{\log ^5\left (\frac {x}{5}\right )}dx}{2500}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (5 \log \left (\frac {x}{5}\right )-4\right ) \log ^4(x) x^4}{\log ^5\left (\frac {x}{5}\right )}+\frac {4 e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \log ^3(x) x^4}{\log ^4\left (\frac {x}{5}\right )}\right )dx}{2500}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \int \frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \log ^4(x)}{\log ^4\left (\frac {x}{5}\right )}dx-4 \int \frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \log ^4(x)}{\log ^5\left (\frac {x}{5}\right )}dx+4 \int \frac {e^{-\frac {30000 \log ^4\left (\frac {x}{5}\right )-x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \log ^3(x)}{\log ^4\left (\frac {x}{5}\right )}dx}{2500}\) |
Input:
Int[(E^((-30000*Log[x/5]^4 + x^5*Log[x]^4)/(2500*Log[x/5]^4))*(4*x^4*Log[x /5]*Log[x]^3 + (-4*x^4 + 5*x^4*Log[x/5])*Log[x]^4))/(2500*Log[x/5]^5),x]
Output:
$Aborted
Time = 18.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {x^{5} \ln \left (x \right )^{4}-30000 \ln \left (\frac {x}{5}\right )^{4}}{2500 \ln \left (\frac {x}{5}\right )^{4}}}\) | \(29\) |
| risch | \({\mathrm e}^{-\frac {-x^{5} \ln \left (x \right )^{4}+30000 \ln \left (5\right )^{4}-120000 \ln \left (5\right )^{3} \ln \left (x \right )+180000 \ln \left (5\right )^{2} \ln \left (x \right )^{2}-120000 \ln \left (5\right ) \ln \left (x \right )^{3}+30000 \ln \left (x \right )^{4}}{2500 \left (-\ln \left (x \right )+\ln \left (5\right )\right )^{4}}}\) | \(61\) |
Input:
int(1/2500*((5*x^4*ln(1/5*x)-4*x^4)*ln(x)^4+4*x^4*ln(1/5*x)*ln(x)^3)*exp(1 /10000*(x^5*ln(x)^4-30000*ln(1/5*x)^4)/ln(1/5*x)^4)^4/ln(1/5*x)^5,x,method =_RETURNVERBOSE)
Output:
exp(1/10000*(x^5*ln(x)^4-30000*ln(1/5*x)^4)/ln(1/5*x)^4)^4
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (5\right )^{4} + 4 \, x^{5} \log \left (5\right )^{3} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{5} \log \left (5\right )^{2} \log \left (\frac {1}{5} \, x\right )^{2} + 4 \, x^{5} \log \left (5\right ) \log \left (\frac {1}{5} \, x\right )^{3} + {\left (x^{5} - 30000\right )} \log \left (\frac {1}{5} \, x\right )^{4}}{2500 \, \log \left (\frac {1}{5} \, x\right )^{4}}\right )} \] Input:
integrate(1/2500*((5*x^4*log(1/5*x)-4*x^4)*log(x)^4+4*x^4*log(1/5*x)*log(x )^3)*exp(1/10000*(x^5*log(x)^4-30000*log(1/5*x)^4)/log(1/5*x)^4)^4/log(1/5 *x)^5,x, algorithm="fricas")
Output:
e^(1/2500*(x^5*log(5)^4 + 4*x^5*log(5)^3*log(1/5*x) + 6*x^5*log(5)^2*log(1 /5*x)^2 + 4*x^5*log(5)*log(1/5*x)^3 + (x^5 - 30000)*log(1/5*x)^4)/log(1/5* x)^4)
Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\frac {4 \left (\frac {x^{5} \log {\left (x \right )}^{4}}{10000} - 3 \left (\log {\left (x \right )} - \log {\left (5 \right )}\right )^{4}\right )}{\left (\log {\left (x \right )} - \log {\left (5 \right )}\right )^{4}}} \] Input:
integrate(1/2500*((5*x**4*ln(1/5*x)-4*x**4)*ln(x)**4+4*x**4*ln(1/5*x)*ln(x )**3)*exp(1/10000*(x**5*ln(x)**4-30000*ln(1/5*x)**4)/ln(1/5*x)**4)**4/ln(1 /5*x)**5,x)
Output:
exp(4*(x**5*log(x)**4/10000 - 3*(log(x) - log(5))**4)/(log(x) - log(5))**4 )
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (x\right )^{4}}{2500 \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} \log \left (x\right ) + 6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 4 \, \log \left (5\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}} - 12\right )} \] Input:
integrate(1/2500*((5*x^4*log(1/5*x)-4*x^4)*log(x)^4+4*x^4*log(1/5*x)*log(x )^3)*exp(1/10000*(x^5*log(x)^4-30000*log(1/5*x)^4)/log(1/5*x)^4)^4/log(1/5 *x)^5,x, algorithm="maxima")
Output:
e^(1/2500*x^5*log(x)^4/(log(5)^4 - 4*log(5)^3*log(x) + 6*log(5)^2*log(x)^2 - 4*log(5)*log(x)^3 + log(x)^4) - 12)
Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (x\right )^{4}}{2500 \, \log \left (\frac {1}{5} \, x\right )^{4}} - 12\right )} \] Input:
integrate(1/2500*((5*x^4*log(1/5*x)-4*x^4)*log(x)^4+4*x^4*log(1/5*x)*log(x )^3)*exp(1/10000*(x^5*log(x)^4-30000*log(1/5*x)^4)/log(1/5*x)^4)^4/log(1/5 *x)^5,x, algorithm="giac")
Output:
e^(1/2500*x^5*log(x)^4/log(1/5*x)^4 - 12)
Time = 2.98 (sec) , antiderivative size = 277, normalized size of antiderivative = 12.04 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=5^{\frac {48\,{\ln \left (x\right )}^3}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,x^{\frac {48\,{\ln \left (5\right )}^3}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{-\frac {72\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{\frac {x^5\,{\ln \left (x\right )}^4}{2500\,\left ({\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4\right )}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (x\right )}^4}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (5\right )}^4}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}} \] Input:
int((exp((4*((x^5*log(x)^4)/10000 - 3*log(x/5)^4))/log(x/5)^4)*(log(x)^4*( 5*x^4*log(x/5) - 4*x^4) + 4*x^4*log(x/5)*log(x)^3))/(2500*log(x/5)^5),x)
Output:
5^((48*log(x)^3)/(log(x)^4 - 4*log(5)^3*log(x) - 4*log(5)*log(x)^3 + 6*log (5)^2*log(x)^2 + log(5)^4))*x^((48*log(5)^3)/(log(x)^4 - 4*log(5)^3*log(x) - 4*log(5)*log(x)^3 + 6*log(5)^2*log(x)^2 + log(5)^4))*exp(-(72*log(5)^2* log(x)^2)/(log(x)^4 - 4*log(5)^3*log(x) - 4*log(5)*log(x)^3 + 6*log(5)^2*l og(x)^2 + log(5)^4))*exp((x^5*log(x)^4)/(2500*(log(x)^4 - 4*log(5)^3*log(x ) - 4*log(5)*log(x)^3 + 6*log(5)^2*log(x)^2 + log(5)^4)))*exp(-(12*log(x)^ 4)/(log(x)^4 - 4*log(5)^3*log(x) - 4*log(5)*log(x)^3 + 6*log(5)^2*log(x)^2 + log(5)^4))*exp(-(12*log(5)^4)/(log(x)^4 - 4*log(5)^3*log(x) - 4*log(5)* log(x)^3 + 6*log(5)^2*log(x)^2 + log(5)^4))
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=\frac {e^{\frac {\mathrm {log}\left (x \right )^{4} x^{5}}{2500 \mathrm {log}\left (\frac {x}{5}\right )^{4}}}}{e^{12}} \] Input:
int(1/2500*((5*x^4*log(1/5*x)-4*x^4)*log(x)^4+4*x^4*log(1/5*x)*log(x)^3)*e xp(1/10000*(x^5*log(x)^4-30000*log(1/5*x)^4)/log(1/5*x)^4)^4/log(1/5*x)^5, x)
Output:
e**((log(x)**4*x**5)/(2500*log(x/5)**4))/e**12