Integrand size = 239, antiderivative size = 30 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x (2+x)-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \]
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=-\frac {x}{-2 x-x^2+\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \]
Integrate[(4*E^5 - 5*x + (-(E^5*x^2) + x^3)*Log[9*E^5*x^4 - 9*x^5] + (-E^5 + x)*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5])/((-4*x^3 - 4*x ^4 - x^5 + E^5*(4*x^2 + 4*x^3 + x^4))*Log[9*E^5*x^4 - 9*x^5] + (4*x^2 + 2* x^3 + E^5*(-4*x - 2*x^2))*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5 ]/5] + (E^5 - x)*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5]^2),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.
| \(\displaystyle \int \frac {\left (x-e^5\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (x^3-e^5 x^2\right ) \log \left (9 e^5 x^4-9 x^5\right )-5 x+4 e^5}{\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (2 x^3+4 x^2+e^5 \left (-2 x^2-4 x\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (-x^5-4 x^4-4 x^3+e^5 \left (x^4+4 x^3+4 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )+x^2\right )-5 x+4 e^5}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (x (x+2)-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )+x^2+2 x}+\frac {-2 e^5 x \log \left (9 \left (e^5-x\right ) x^4\right )+2 x^3 \log \left (9 \left (e^5-x\right ) x^4\right )+2 \left (1-e^5\right ) x^2 \log \left (9 \left (e^5-x\right ) x^4\right )-5 x+4 e^5}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )+x^2+2 x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e^5 \left (1-e^5\right ) \int \frac {1}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-2 e^{10} \int \frac {1}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx+2 e^5 \int \frac {1}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx+2 e^{10} \left (1-e^5\right ) \int \frac {1}{\left (e^5-x\right ) \left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx+2 e^{15} \int \frac {1}{\left (e^5-x\right ) \left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-2 e^{10} \int \frac {1}{\left (e^5-x\right ) \left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-2 \left (1-e^5\right ) \int \frac {x}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-2 e^5 \int \frac {x}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-2 \int \frac {x^2}{\left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx+5 \int \frac {1}{\log \left (9 \left (e^5-x\right ) x^4\right ) \left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx-e^5 \int \frac {1}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}dx+\int \frac {1}{x^2+2 x-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )}dx\) |
Int[(4*E^5 - 5*x + (-(E^5*x^2) + x^3)*Log[9*E^5*x^4 - 9*x^5] + (-E^5 + x)* Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5])/((-4*x^3 - 4*x^4 - x ^5 + E^5*(4*x^2 + 4*x^3 + x^4))*Log[9*E^5*x^4 - 9*x^5] + (4*x^2 + 2*x^3 + E^5*(-4*x - 2*x^2))*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5] + (E^5 - x)*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5]^2),x]
3.28.65.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 4.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\frac {x}{x^{2}+2 x -\ln \left (\frac {\ln \left (9 x^{4} \left ({\mathrm e}^{5}-x \right )\right )}{5}\right )}\) | \(29\) |
int(((-exp(5)+x)*ln(9*x^4*exp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))+(-x ^2*exp(5)+x^3)*ln(9*x^4*exp(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)*ln(9*x^4*e xp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)*exp(5)+2*x^3+4 *x^2)*ln(9*x^4*exp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))+((x^4+4*x^3+4* x^2)*exp(5)-x^5-4*x^4-4*x^3)*ln(9*x^4*exp(5)-9*x^5)),x,method=_RETURNVERBO SE)
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x - \log \left (\frac {1}{5} \, \log \left (-9 \, x^{5} + 9 \, x^{4} e^{5}\right )\right )} \]
integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9* x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*exp(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)* log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)*e xp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5) )+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, a lgorithm=\
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=- \frac {x}{- x^{2} - 2 x + \log {\left (\frac {\log {\left (- 9 x^{5} + 9 x^{4} e^{5} \right )}}{5} \right )}} \]
integrate(((-exp(5)+x)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp(5)-9* x**5))+(-x**2*exp(5)+x**3)*ln(9*x**4*exp(5)-9*x**5)+4*exp(5)-5*x)/((exp(5) -x)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp(5)-9*x**5))**2+((-2*x**2 -4*x)*exp(5)+2*x**3+4*x**2)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp( 5)-9*x**5))+((x**4+4*x**3+4*x**2)*exp(5)-x**5-4*x**4-4*x**3)*ln(9*x**4*exp (5)-9*x**5)),x)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x + \log \left (5\right ) - \log \left (i \, \pi + 2 \, \log \left (3\right ) + \log \left (x - e^{5}\right ) + 4 \, \log \left (x\right )\right )} \]
integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9* x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*exp(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)* log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)*e xp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5) )+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, a lgorithm=\
Time = 1.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x + \log \left (5\right ) - \log \left (\log \left (-9 \, x^{5} + 9 \, x^{4} e^{5}\right )\right )} \]
integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9* x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*exp(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)* log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)*e xp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5) )+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, a lgorithm=\
Timed out. \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\text {Hanged} \]
int((5*x - 4*exp(5) + log(9*x^4*exp(5) - 9*x^5)*(x^2*exp(5) - x^3) - log(l og(9*x^4*exp(5) - 9*x^5)/5)*log(9*x^4*exp(5) - 9*x^5)*(x - exp(5)))/(log(9 *x^4*exp(5) - 9*x^5)*(4*x^3 - exp(5)*(4*x^2 + 4*x^3 + x^4) + 4*x^4 + x^5) + log(log(9*x^4*exp(5) - 9*x^5)/5)^2*log(9*x^4*exp(5) - 9*x^5)*(x - exp(5) ) - log(log(9*x^4*exp(5) - 9*x^5)/5)*log(9*x^4*exp(5) - 9*x^5)*(4*x^2 - ex p(5)*(4*x + 2*x^2) + 2*x^3)),x)