\(\int \frac {e^{\frac {-30000 \log ^4(\frac {x}{5})+x^5 \log ^4(x)}{2500 \log ^4(\frac {x}{5})}} (4 x^4 \log (\frac {x}{5}) \log ^3(x)+(-4 x^4+5 x^4 \log (\frac {x}{5})) \log ^4(x))}{2500 \log ^5(\frac {x}{5})} \, dx\) [6671]

Optimal result

Integrand size = 83, antiderivative size = 23 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \]

[Out]

exp(1/10000*x^5/ln(1/5*x)^4*ln(x)^4-3)^4

Rubi [F]

\[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=\int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx \]

[In]

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]

[Out]

Defer[Int][(E^(-12 + (x^5*Log[x]^4)/(2500*Log[x/5]^4))*x^4*Log[x]^3)/Log[x/5]^4, x]/625 - Defer[Int][(E^(-12 +
(x^5*Log[x]^4)/(2500*Log[x/5]^4))*x^4*Log[x]^4)/Log[x/5]^5, x]/625 + Defer[Int][(E^(-12 + (x^5*Log[x]^4)/(250
0*Log[x/5]^4))*x^4*Log[x]^4)/Log[x/5]^4, x]/500

Rubi steps \begin{align*} \text {integral}& = \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} \\ & = \frac {\int \frac {e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \log ^3(x) \left (-4 \log (x)+\log \left (\frac {x}{5}\right ) (4+5 \log (x))\right )}{\log ^5\left (\frac {x}{5}\right )} \, dx}{2500} \\ & = \frac {\int \left (\frac {4 e^{-12+\frac {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 )}+\frac {e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \left (-4+5 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)}{\log ^5\left (\frac {x}{5}\right )}\right ) \, dx}{2500} \\ & = \frac {\int \frac {e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} x^4 \left (-4+5 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)}{\log ^5\left (\frac {x}{5}\right )} \, dx}{2500}+\frac {1}{625} \int \frac {e^{-12+\frac {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 \\ & = \frac {\int \left (-\frac {4 e^{-12+\frac {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 )}+\frac {5 e^{-12+\frac {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 )}\right ) \, dx}{2500}+\frac {1}{625} \int \frac {e^{-12+\frac {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 \\ & = \frac {1}{625} \int \frac {e^{-12+\frac {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-\frac {1}{625} \int \frac {e^{-12+\frac {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+\frac {1}{500} \int \frac {e^{-12+\frac {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 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{-12+\frac {x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \]

[In]

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]

[Out]

E^(-12 + (x^5*Log[x]^4)/(2500*Log[x/5]^4))

Maple [A] (verified)

Time = 20.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30

method	result	size
parallelrisch	\({\mathrm e}^{-\frac {-x^{5} \ln \left (x \right )^{4}+30000 \ln \left (\frac {x}{5}\right )^{4}}{2500 \ln \left (\frac {x}{5}\right )^{4}}}\)	\(30\)
risch	\({\mathrm e}^{-\frac {-x^{5} \ln \left (x \right )^{4}+30000 \ln \left (5\right )^{4}-120000 \ln \left (x \right ) \ln \left (5\right )^{3}+180000 \ln \left (x \right )^{2} \ln \left (5\right )^{2}-120000 \ln \left (5\right ) \ln \left (x \right )^{3}+30000 \ln \left (x \right )^{4}}{2500 \left (\ln \left (5\right )-\ln \left (x \right )\right )^{4}}}\)	\(61\)

[In]

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)

[Out]

exp(-1/10000*(-x^5*ln(x)^4+30000*ln(1/5*x)^4)/ln(1/5*x)^4)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (18) = 36\).

Time = 0.52 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (5\right )^{4} + 4 \, x^{5} \log \left (5\right )^{3} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{5} \log \left (5\right )^{2} \log \left (\frac {1}{5} \, x\right )^{2} + 4 \, x^{5} \log \left (5\right ) \log \left (\frac {1}{5} \, x\right )^{3} + {\left (x^{5} - 30000\right )} \log \left (\frac {1}{5} \, x\right )^{4}}{2500 \, \log \left (\frac {1}{5} \, x\right )^{4}}\right )} \]

[In]

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")

[Out]

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)

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\frac {4 \left (\frac {x^{5} \log {\left (x \right )}^{4}}{10000} - 3 \left (\log {\left (x \right )} - \log {\left (5 \right )}\right )^{4}\right )}{\left (\log {\left (x \right )} - \log {\left (5 \right )}\right )^{4}}} \]

[In]

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)**4-300
00*ln(1/5*x)**4)/ln(1/5*x)**4)**4/ln(1/5*x)**5,x)

[Out]

exp(4*(x**5*log(x)**4/10000 - 3*(log(x) - log(5))**4)/(log(x) - log(5))**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).

Time = 0.44 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (x\right )^{4}}{2500 \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} \log \left (x\right ) + 6 \, \log \left (5\right )^{2} \log \left (x\right )^{2} - 4 \, \log \left (5\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}} - 12\right )} \]

[In]

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")

[Out]

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) - 1
2)

Giac [A] (verification not implemented)

none

Time = 0.89 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=e^{\left (\frac {x^{5} \log \left (x\right )^{4}}{2500 \, \log \left (\frac {1}{5} \, x\right )^{4}} - 12\right )} \]

[In]

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")

[Out]

e^(1/2500*x^5*log(x)^4/log(1/5*x)^4 - 12)

Mupad [B] (verification not implemented)

Time = 14.47 (sec) , antiderivative size = 277, normalized size of antiderivative = 12.04 \[ \int \frac {e^{\frac {-30000 \log ^4\left (\frac {x}{5}\right )+x^5 \log ^4(x)}{2500 \log ^4\left (\frac {x}{5}\right )}} \left (4 x^4 \log \left (\frac {x}{5}\right ) \log ^3(x)+\left (-4 x^4+5 x^4 \log \left (\frac {x}{5}\right )\right ) \log ^4(x)\right )}{2500 \log ^5\left (\frac {x}{5}\right )} \, dx=5^{\frac {48\,{\ln \left (x\right )}^3}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,x^{\frac {48\,{\ln \left (5\right )}^3}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{-\frac {72\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{\frac {x^5\,{\ln \left (x\right )}^4}{2500\,\left ({\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4\right )}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (x\right )}^4}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (5\right )}^4}{{\ln \left (x\right )}^4-4\,\ln \left (5\right )\,{\ln \left (x\right )}^3+6\,{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2-4\,{\ln \left (5\right )}^3\,\ln \left (x\right )+{\ln \left (5\right )}^4}} \]

[In]

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)

[Out]

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*l
og(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*log(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))