Integrand size = 301, antiderivative size = 36 \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=\left (1+e^{\frac {1}{5} x^2 \left (-x+\frac {e^5}{3 \log \left (\log \left (5+\frac {x}{4}\right )\right )}\right )}+x\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(36)=72\).
Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.25 \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=e^{-\frac {2 x^3}{5}} \left (e^{\frac {2 e^5 x^2}{15 \log \left (\log \left (5+\frac {x}{4}\right )\right )}}+2 e^{\frac {1}{15} x^2 \left (3 x+\frac {e^5}{\log \left (\log \left (5+\frac {x}{4}\right )\right )}\right )} (1+x)+e^{\frac {2 x^3}{5}} x (2+x)\right ) \]
Integrate[((600 + 630*x + 30*x^2)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2 + E^((2*(E^5*x^2 - 3*x^3*Log[Log[(20 + x)/4]]))/(15*Log[Log[(20 + x)/4]]))* (-2*E^5*x^2 + E^5*(80*x + 4*x^2)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]] + (- 360*x^2 - 18*x^3)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2) + E^((E^5*x^2 - 3*x^3*Log[Log[(20 + x)/4]])/(15*Log[Log[(20 + x)/4]]))*(E^5*(-2*x^2 - 2*x^ 3) + E^5*(80*x + 84*x^2 + 4*x^3)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]] + (6 00 + 30*x - 360*x^2 - 378*x^3 - 18*x^4)*Log[(20 + x)/4]*Log[Log[(20 + x)/4 ]]^2))/((300 + 15*x)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2),x]
(E^((2*E^5*x^2)/(15*Log[Log[5 + x/4]])) + 2*E^((x^2*(3*x + E^5/Log[Log[5 + x/4]]))/15)*(1 + x) + E^((2*x^3)/5)*x*(2 + x))/E^((2*x^3)/5)
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(36)=72\).
Time = 12.41 (sec) , antiderivative size = 314, normalized size of antiderivative = 8.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {\left (-2 e^5 x^2+e^5 \left (4 x^2+80 x\right ) \log \left (\frac {x+20}{4}\right ) \log \left (\log \left (\frac {x+20}{4}\right )\right )+\left (-18 x^3-360 x^2\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )\right ) \exp \left (\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {x+20}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {x+20}{4}\right )\right )}\right )+\left (e^5 \left (-2 x^3-2 x^2\right )+e^5 \left (4 x^3+84 x^2+80 x\right ) \log \left (\frac {x+20}{4}\right ) \log \left (\log \left (\frac {x+20}{4}\right )\right )+\left (-18 x^4-378 x^3-360 x^2+30 x+600\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )\right ) \exp \left (\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {x+20}{4}\right )\right )}{15 \log \left (\log \left (\frac {x+20}{4}\right )\right )}\right )+\left (30 x^2+630 x+600\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )}{(15 x+300) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-2 e^5 x^2+e^5 \left (4 x^2+80 x\right ) \log \left (\frac {x+20}{4}\right ) \log \left (\log \left (\frac {x+20}{4}\right )\right )+\left (-18 x^3-360 x^2\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )\right ) \exp \left (\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {x+20}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {x+20}{4}\right )\right )}\right )+\left (e^5 \left (-2 x^3-2 x^2\right )+e^5 \left (4 x^3+84 x^2+80 x\right ) \log \left (\frac {x+20}{4}\right ) \log \left (\log \left (\frac {x+20}{4}\right )\right )+\left (-18 x^4-378 x^3-360 x^2+30 x+600\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )\right ) \exp \left (\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {x+20}{4}\right )\right )}{15 \log \left (\log \left (\frac {x+20}{4}\right )\right )}\right )+\left (30 x^2+630 x+600\right ) \log \left (\frac {x+20}{4}\right ) \log ^2\left (\log \left (\frac {x+20}{4}\right )\right )}{(15 x+300) \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x e^{-\frac {2}{15} x^2 \left (3 x-\frac {e^5}{\log \left (\log \left (\frac {x}{4}+5\right )\right )}\right )} \left (9 x^2 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )+e^5 x+180 x \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-2 e^5 x \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )-40 e^5 \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )\right )}{15 (x+20) \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )}-\frac {2 e^{-\frac {1}{15} x^2 \left (3 x-\frac {e^5}{\log \left (\log \left (\frac {x}{4}+5\right )\right )}\right )} \left (9 x^4 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )+e^5 x^3+189 x^3 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-2 e^5 x^3 \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )+e^5 x^2+180 x^2 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-42 e^5 x^2 \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )-15 x \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-300 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-40 e^5 x \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )\right )}{15 (x+20) \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )}+2 (x+1)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^{-\frac {2}{15} x^2 \left (3 x-\frac {e^5}{\log \left (\log \left (\frac {x}{4}+5\right )\right )}\right )}+\frac {2 e^{-\frac {1}{15} x^2 \left (3 x-\frac {e^5}{\log \left (\log \left (\frac {x}{4}+5\right )\right )}\right )} \left (9 x^4 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )+e^5 x^3+189 x^3 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-2 e^5 x^3 \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )+e^5 x^2+180 x^2 \log \left (\frac {x}{4}+5\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )-42 e^5 x^2 \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )-40 e^5 x \log \left (\frac {x}{4}+5\right ) \log \left (\log \left (\frac {x}{4}+5\right )\right )\right )}{(x+20) \log \left (\frac {x}{4}+5\right ) \left (x^2 \left (\frac {e^5}{(x+20) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right ) \log \left (\frac {x}{4}+5\right )}+3\right )+2 x \left (3 x-\frac {e^5}{\log \left (\log \left (\frac {x}{4}+5\right )\right )}\right )\right ) \log ^2\left (\log \left (\frac {x}{4}+5\right )\right )}+(x+1)^2\) |
Int[((600 + 630*x + 30*x^2)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2 + E^((2 *(E^5*x^2 - 3*x^3*Log[Log[(20 + x)/4]]))/(15*Log[Log[(20 + x)/4]]))*(-2*E^ 5*x^2 + E^5*(80*x + 4*x^2)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]] + (-360*x^ 2 - 18*x^3)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2) + E^((E^5*x^2 - 3*x^3* Log[Log[(20 + x)/4]])/(15*Log[Log[(20 + x)/4]]))*(E^5*(-2*x^2 - 2*x^3) + E ^5*(80*x + 84*x^2 + 4*x^3)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]] + (600 + 3 0*x - 360*x^2 - 378*x^3 - 18*x^4)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2)) /((300 + 15*x)*Log[(20 + x)/4]*Log[Log[(20 + x)/4]]^2),x]
E^((-2*x^2*(3*x - E^5/Log[Log[5 + x/4]]))/15) + (1 + x)^2 + (2*(E^5*x^2 + E^5*x^3 - 40*E^5*x*Log[5 + x/4]*Log[Log[5 + x/4]] - 42*E^5*x^2*Log[5 + x/4 ]*Log[Log[5 + x/4]] - 2*E^5*x^3*Log[5 + x/4]*Log[Log[5 + x/4]] + 180*x^2*L og[5 + x/4]*Log[Log[5 + x/4]]^2 + 189*x^3*Log[5 + x/4]*Log[Log[5 + x/4]]^2 + 9*x^4*Log[5 + x/4]*Log[Log[5 + x/4]]^2))/(E^((x^2*(3*x - E^5/Log[Log[5 + x/4]]))/15)*(20 + x)*Log[5 + x/4]*(x^2*(3 + E^5/((20 + x)*Log[5 + x/4]*L og[Log[5 + x/4]]^2)) + 2*x*(3*x - E^5/Log[Log[5 + x/4]]))*Log[Log[5 + x/4] ]^2)
3.6.56.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(28)=56\).
Time = 97.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(x^{2}+{\mathrm e}^{\frac {2 x^{2} \left (-3 \ln \left (\ln \left (5+\frac {x}{4}\right )\right ) x +{\mathrm e}^{5}\right )}{15 \ln \left (\ln \left (5+\frac {x}{4}\right )\right )}}+2 x +\left (2+2 x \right ) {\mathrm e}^{\frac {x^{2} \left (-3 \ln \left (\ln \left (5+\frac {x}{4}\right )\right ) x +{\mathrm e}^{5}\right )}{15 \ln \left (\ln \left (5+\frac {x}{4}\right )\right )}}\) | \(70\) |
| parallelrisch | \(-480+x^{2}+2 \,{\mathrm e}^{\frac {x^{2} \left (-3 \ln \left (\ln \left (5+\frac {x}{4}\right )\right ) x +{\mathrm e}^{5}\right )}{15 \ln \left (\ln \left (5+\frac {x}{4}\right )\right )}} x +{\mathrm e}^{\frac {2 x^{2} \left (-3 \ln \left (\ln \left (5+\frac {x}{4}\right )\right ) x +{\mathrm e}^{5}\right )}{15 \ln \left (\ln \left (5+\frac {x}{4}\right )\right )}}+2 x +2 \,{\mathrm e}^{\frac {x^{2} \left (-3 \ln \left (\ln \left (5+\frac {x}{4}\right )\right ) x +{\mathrm e}^{5}\right )}{15 \ln \left (\ln \left (5+\frac {x}{4}\right )\right )}}\) | \(100\) |
int((((-18*x^3-360*x^2)*ln(5+1/4*x)*ln(ln(5+1/4*x))^2+(4*x^2+80*x)*exp(5)* ln(5+1/4*x)*ln(ln(5+1/4*x))-2*x^2*exp(5))*exp(1/15*(-3*x^3*ln(ln(5+1/4*x)) +x^2*exp(5))/ln(ln(5+1/4*x)))^2+((-18*x^4-378*x^3-360*x^2+30*x+600)*ln(5+1 /4*x)*ln(ln(5+1/4*x))^2+(4*x^3+84*x^2+80*x)*exp(5)*ln(5+1/4*x)*ln(ln(5+1/4 *x))+(-2*x^3-2*x^2)*exp(5))*exp(1/15*(-3*x^3*ln(ln(5+1/4*x))+x^2*exp(5))/l n(ln(5+1/4*x)))+(30*x^2+630*x+600)*ln(5+1/4*x)*ln(ln(5+1/4*x))^2)/(15*x+30 0)/ln(5+1/4*x)/ln(ln(5+1/4*x))^2,x,method=_RETURNVERBOSE)
x^2+exp(2/15*x^2*(-3*ln(ln(5+1/4*x))*x+exp(5))/ln(ln(5+1/4*x)))+2*x+(2+2*x )*exp(1/15*x^2*(-3*ln(ln(5+1/4*x))*x+exp(5))/ln(ln(5+1/4*x)))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.11 \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=x^{2} + 2 \, {\left (x + 1\right )} e^{\left (-\frac {3 \, x^{3} \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - x^{2} e^{5}}{15 \, \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )}\right )} + 2 \, x + e^{\left (-\frac {2 \, {\left (3 \, x^{3} \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - x^{2} e^{5}\right )}}{15 \, \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )}\right )} \]
integrate((((-18*x^3-360*x^2)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^2+80*x )*exp(5)*log(5+1/4*x)*log(log(5+1/4*x))-2*x^2*exp(5))*exp(1/15*(-3*x^3*log (log(5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))^2+((-18*x^4-378*x^3-360*x^2+ 30*x+600)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^3+84*x^2+80*x)*exp(5)*log( 5+1/4*x)*log(log(5+1/4*x))+(-2*x^3-2*x^2)*exp(5))*exp(1/15*(-3*x^3*log(log (5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))+(30*x^2+630*x+600)*log(5+1/4*x)* log(log(5+1/4*x))^2)/(15*x+300)/log(5+1/4*x)/log(log(5+1/4*x))^2,x, algori thm=\
x^2 + 2*(x + 1)*e^(-1/15*(3*x^3*log(log(1/4*x + 5)) - x^2*e^5)/log(log(1/4 *x + 5))) + 2*x + e^(-2/15*(3*x^3*log(log(1/4*x + 5)) - x^2*e^5)/log(log(1 /4*x + 5)))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (26) = 52\).
Time = 1.97 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=x^{2} + 2 x + \left (2 x + 2\right ) e^{\frac {- \frac {x^{3} \log {\left (\log {\left (\frac {x}{4} + 5 \right )} \right )}}{5} + \frac {x^{2} e^{5}}{15}}{\log {\left (\log {\left (\frac {x}{4} + 5 \right )} \right )}}} + e^{\frac {2 \left (- \frac {x^{3} \log {\left (\log {\left (\frac {x}{4} + 5 \right )} \right )}}{5} + \frac {x^{2} e^{5}}{15}\right )}{\log {\left (\log {\left (\frac {x}{4} + 5 \right )} \right )}}} \]
integrate((((-18*x**3-360*x**2)*ln(5+1/4*x)*ln(ln(5+1/4*x))**2+(4*x**2+80* x)*exp(5)*ln(5+1/4*x)*ln(ln(5+1/4*x))-2*x**2*exp(5))*exp(1/15*(-3*x**3*ln( ln(5+1/4*x))+x**2*exp(5))/ln(ln(5+1/4*x)))**2+((-18*x**4-378*x**3-360*x**2 +30*x+600)*ln(5+1/4*x)*ln(ln(5+1/4*x))**2+(4*x**3+84*x**2+80*x)*exp(5)*ln( 5+1/4*x)*ln(ln(5+1/4*x))+(-2*x**3-2*x**2)*exp(5))*exp(1/15*(-3*x**3*ln(ln( 5+1/4*x))+x**2*exp(5))/ln(ln(5+1/4*x)))+(30*x**2+630*x+600)*ln(5+1/4*x)*ln (ln(5+1/4*x))**2)/(15*x+300)/ln(5+1/4*x)/ln(ln(5+1/4*x))**2,x)
x**2 + 2*x + (2*x + 2)*exp((-x**3*log(log(x/4 + 5))/5 + x**2*exp(5)/15)/lo g(log(x/4 + 5))) + exp(2*(-x**3*log(log(x/4 + 5))/5 + x**2*exp(5)/15)/log( log(x/4 + 5)))
Exception generated. \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate((((-18*x^3-360*x^2)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^2+80*x )*exp(5)*log(5+1/4*x)*log(log(5+1/4*x))-2*x^2*exp(5))*exp(1/15*(-3*x^3*log (log(5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))^2+((-18*x^4-378*x^3-360*x^2+ 30*x+600)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^3+84*x^2+80*x)*exp(5)*log( 5+1/4*x)*log(log(5+1/4*x))+(-2*x^3-2*x^2)*exp(5))*exp(1/15*(-3*x^3*log(log (5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))+(30*x^2+630*x+600)*log(5+1/4*x)* log(log(5+1/4*x))^2)/(15*x+300)/log(5+1/4*x)/log(log(5+1/4*x))^2,x, algori thm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=\int { \frac {2 \, {\left (15 \, {\left (x^{2} + 21 \, x + 20\right )} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )^{2} + {\left (2 \, {\left (x^{3} + 21 \, x^{2} + 20 \, x\right )} e^{5} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - 3 \, {\left (3 \, x^{4} + 63 \, x^{3} + 60 \, x^{2} - 5 \, x - 100\right )} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )^{2} - {\left (x^{3} + x^{2}\right )} e^{5}\right )} e^{\left (-\frac {3 \, x^{3} \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - x^{2} e^{5}}{15 \, \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )}\right )} + {\left (2 \, {\left (x^{2} + 20 \, x\right )} e^{5} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - 9 \, {\left (x^{3} + 20 \, x^{2}\right )} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )^{2} - x^{2} e^{5}\right )} e^{\left (-\frac {2 \, {\left (3 \, x^{3} \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right ) - x^{2} e^{5}\right )}}{15 \, \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )}\right )}\right )}}{15 \, {\left (x + 20\right )} \log \left (\frac {1}{4} \, x + 5\right ) \log \left (\log \left (\frac {1}{4} \, x + 5\right )\right )^{2}} \,d x } \]
integrate((((-18*x^3-360*x^2)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^2+80*x )*exp(5)*log(5+1/4*x)*log(log(5+1/4*x))-2*x^2*exp(5))*exp(1/15*(-3*x^3*log (log(5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))^2+((-18*x^4-378*x^3-360*x^2+ 30*x+600)*log(5+1/4*x)*log(log(5+1/4*x))^2+(4*x^3+84*x^2+80*x)*exp(5)*log( 5+1/4*x)*log(log(5+1/4*x))+(-2*x^3-2*x^2)*exp(5))*exp(1/15*(-3*x^3*log(log (5+1/4*x))+x^2*exp(5))/log(log(5+1/4*x)))+(30*x^2+630*x+600)*log(5+1/4*x)* log(log(5+1/4*x))^2)/(15*x+300)/log(5+1/4*x)/log(log(5+1/4*x))^2,x, algori thm=\
Time = 8.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.25 \[ \int \frac {\left (600+630 x+30 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )+e^{\frac {2 \left (e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (-2 e^5 x^2+e^5 \left (80 x+4 x^2\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (-360 x^2-18 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )+e^{\frac {e^5 x^2-3 x^3 \log \left (\log \left (\frac {20+x}{4}\right )\right )}{15 \log \left (\log \left (\frac {20+x}{4}\right )\right )}} \left (e^5 \left (-2 x^2-2 x^3\right )+e^5 \left (80 x+84 x^2+4 x^3\right ) \log \left (\frac {20+x}{4}\right ) \log \left (\log \left (\frac {20+x}{4}\right )\right )+\left (600+30 x-360 x^2-378 x^3-18 x^4\right ) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )\right )}{(300+15 x) \log \left (\frac {20+x}{4}\right ) \log ^2\left (\log \left (\frac {20+x}{4}\right )\right )} \, dx=2\,x+2\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^5}{15\,\ln \left (\ln \left (\frac {x}{4}+5\right )\right )}-\frac {x^3}{5}}+{\mathrm {e}}^{\frac {2\,x^2\,{\mathrm {e}}^5}{15\,\ln \left (\ln \left (\frac {x}{4}+5\right )\right )}-\frac {2\,x^3}{5}}+x^2+2\,x\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^5}{15\,\ln \left (\ln \left (\frac {x}{4}+5\right )\right )}-\frac {x^3}{5}} \]
int(-(exp(-((x^3*log(log(x/4 + 5)))/5 - (x^2*exp(5))/15)/log(log(x/4 + 5)) )*(exp(5)*(2*x^2 + 2*x^3) + log(log(x/4 + 5))^2*log(x/4 + 5)*(360*x^2 - 30 *x + 378*x^3 + 18*x^4 - 600) - log(log(x/4 + 5))*exp(5)*log(x/4 + 5)*(80*x + 84*x^2 + 4*x^3)) + exp(-(2*((x^3*log(log(x/4 + 5)))/5 - (x^2*exp(5))/15 ))/log(log(x/4 + 5)))*(2*x^2*exp(5) + log(log(x/4 + 5))^2*log(x/4 + 5)*(36 0*x^2 + 18*x^3) - log(log(x/4 + 5))*exp(5)*log(x/4 + 5)*(80*x + 4*x^2)) - log(log(x/4 + 5))^2*log(x/4 + 5)*(630*x + 30*x^2 + 600))/(log(log(x/4 + 5) )^2*log(x/4 + 5)*(15*x + 300)),x)