Integrand size = 430, antiderivative size = 29 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=-1+x+\frac {5}{-1-x+\log \left (4 \left (2+\log \left (\frac {-e^3+x}{x}\right )\right )\right )} \]
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=x+\frac {5}{-1-x+\log \left (4 \left (2+\log \left (1-\frac {e^3}{x}\right )\right )\right )} \]
Integrate[(-12*x^2 - 4*x^3 - 2*x^4 + E^3*(5 + 12*x + 4*x^2 + 2*x^3) + (-6* x^2 - 2*x^3 - x^4 + E^3*(6*x + 2*x^2 + x^3))*Log[(-E^3 + x)/x] + (4*x^2 + 4*x^3 + E^3*(-4*x - 4*x^2) + (2*x^2 + 2*x^3 + E^3*(-2*x - 2*x^2))*Log[(-E^ 3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]] + (2*E^3*x - 2*x^2 + (E^3*x - x^2) *Log[(-E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]]^2)/(-2*x^2 - 4*x^3 - 2*x^ 4 + E^3*(2*x + 4*x^2 + 2*x^3) + (-x^2 - 2*x^3 - x^4 + E^3*(x + 2*x^2 + x^3 ))*Log[(-E^3 + x)/x] + (4*x^2 + 4*x^3 + E^3*(-4*x - 4*x^2) + (2*x^2 + 2*x^ 3 + E^3*(-2*x - 2*x^2))*Log[(-E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]] + (2*E^3*x - 2*x^2 + (E^3*x - x^2)*Log[(-E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]]^2),x]
Time = 5.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {7239, 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.
| \(\displaystyle \int \frac {-2 x^4-4 x^3-12 x^2+\left (-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {x-e^3}{x}\right )+2 e^3 x\right ) \log ^2\left (4 \log \left (\frac {x-e^3}{x}\right )+8\right )+e^3 \left (2 x^3+4 x^2+12 x+5\right )+\left (4 x^3+4 x^2+e^3 \left (-4 x^2-4 x\right )+\left (2 x^3+2 x^2+e^3 \left (-2 x^2-2 x\right )\right ) \log \left (\frac {x-e^3}{x}\right )\right ) \log \left (4 \log \left (\frac {x-e^3}{x}\right )+8\right )+\left (-x^4-2 x^3-6 x^2+e^3 \left (x^3+2 x^2+6 x\right )\right ) \log \left (\frac {x-e^3}{x}\right )}{-2 x^4-4 x^3-2 x^2+\left (-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {x-e^3}{x}\right )+2 e^3 x\right ) \log ^2\left (4 \log \left (\frac {x-e^3}{x}\right )+8\right )+e^3 \left (2 x^3+4 x^2+2 x\right )+\left (4 x^3+4 x^2+e^3 \left (-4 x^2-4 x\right )+\left (2 x^3+2 x^2+e^3 \left (-2 x^2-2 x\right )\right ) \log \left (\frac {x-e^3}{x}\right )\right ) \log \left (4 \log \left (\frac {x-e^3}{x}\right )+8\right )+\left (-x^4-2 x^3-x^2+e^3 \left (x^3+2 x^2+x\right )\right ) \log \left (\frac {x-e^3}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 x^4-4 \left (1-\frac {e^3}{2}\right ) x^3-12 \left (1-\frac {e^3}{3}\right ) x^2+\left (e^3-x\right ) x \log \left (1-\frac {e^3}{x}\right ) \left (x^2+2 x+\log ^2\left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )-2 (x+1) \log \left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )+6\right )+12 e^3 x+2 \left (e^3-x\right ) x \log ^2\left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )-4 \left (e^3-x\right ) (x+1) x \log \left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )+5 e^3}{\left (e^3-x\right ) x \left (\log \left (1-\frac {e^3}{x}\right )+2\right ) \left (x-\log \left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 \left (-2 x^2+x^2 \left (-\log \left (1-\frac {e^3}{x}\right )\right )+2 e^3 x+e^3 x \log \left (1-\frac {e^3}{x}\right )+e^3\right )}{\left (e^3-x\right ) x \left (\log \left (1-\frac {e^3}{x}\right )+2\right ) \left (x-\log \left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )+1\right )^2}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x-\frac {5}{x-\log \left (4 \left (\log \left (1-\frac {e^3}{x}\right )+2\right )\right )+1}\) |
Int[(-12*x^2 - 4*x^3 - 2*x^4 + E^3*(5 + 12*x + 4*x^2 + 2*x^3) + (-6*x^2 - 2*x^3 - x^4 + E^3*(6*x + 2*x^2 + x^3))*Log[(-E^3 + x)/x] + (4*x^2 + 4*x^3 + E^3*(-4*x - 4*x^2) + (2*x^2 + 2*x^3 + E^3*(-2*x - 2*x^2))*Log[(-E^3 + x) /x])*Log[8 + 4*Log[(-E^3 + x)/x]] + (2*E^3*x - 2*x^2 + (E^3*x - x^2)*Log[( -E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]]^2)/(-2*x^2 - 4*x^3 - 2*x^4 + E^ 3*(2*x + 4*x^2 + 2*x^3) + (-x^2 - 2*x^3 - x^4 + E^3*(x + 2*x^2 + x^3))*Log [(-E^3 + x)/x] + (4*x^2 + 4*x^3 + E^3*(-4*x - 4*x^2) + (2*x^2 + 2*x^3 + E^ 3*(-2*x - 2*x^2))*Log[(-E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]] + (2*E^3 *x - 2*x^2 + (E^3*x - x^2)*Log[(-E^3 + x)/x])*Log[8 + 4*Log[(-E^3 + x)/x]] ^2),x]
3.7.17.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(28)=56\).
Time = 2.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.83
| method | result | size |
| parallelrisch | \(\frac {-5+x +2 x \,{\mathrm e}^{3}+x^{2}+2 \,{\mathrm e}^{3}-2 \ln \left (4 \ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+8\right ) {\mathrm e}^{3}-\ln \left (4 \ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+8\right ) x}{x -\ln \left (4 \ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+8\right )+1}\) | \(82\) |
| default | \(-\frac {{\mathrm e}^{3}}{-\frac {-x +{\mathrm e}^{3}}{x}-1}+\frac {-\frac {5 \left (-x +{\mathrm e}^{3}\right )}{x}-5}{-\frac {2 \ln \left (2\right ) \left (-x +{\mathrm e}^{3}\right )}{x}-\frac {\left (-x +{\mathrm e}^{3}\right ) \ln \left (\ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+2\right )}{x}-2 \ln \left (2\right )+{\mathrm e}^{3}+\frac {-x +{\mathrm e}^{3}}{x}-\ln \left (\ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+2\right )+1}\) | \(112\) |
int((((x*exp(3)-x^2)*ln((x-exp(3))/x)+2*x*exp(3)-2*x^2)*ln(4*ln((x-exp(3)) /x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*ln((x-exp(3))/x)+(-4*x^2-4*x)* exp(3)+4*x^3+4*x^2)*ln(4*ln((x-exp(3))/x)+8)+((x^3+2*x^2+6*x)*exp(3)-x^4-2 *x^3-6*x^2)*ln((x-exp(3))/x)+(2*x^3+4*x^2+12*x+5)*exp(3)-2*x^4-4*x^3-12*x^ 2)/(((x*exp(3)-x^2)*ln((x-exp(3))/x)+2*x*exp(3)-2*x^2)*ln(4*ln((x-exp(3))/ x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*ln((x-exp(3))/x)+(-4*x^2-4*x)*e xp(3)+4*x^3+4*x^2)*ln(4*ln((x-exp(3))/x)+8)+((x^3+2*x^2+x)*exp(3)-x^4-2*x^ 3-x^2)*ln((x-exp(3))/x)+(2*x^3+4*x^2+2*x)*exp(3)-2*x^4-4*x^3-2*x^2),x,meth od=_RETURNVERBOSE)
(-5+x+2*x*exp(3)+x^2+2*exp(3)-2*ln(4*ln(-(-x+exp(3))/x)+8)*exp(3)-ln(4*ln( -(-x+exp(3))/x)+8)*x)/(x-ln(4*ln(-(-x+exp(3))/x)+8)+1)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=\frac {x^{2} - x \log \left (4 \, \log \left (\frac {x - e^{3}}{x}\right ) + 8\right ) + x - 5}{x - \log \left (4 \, \log \left (\frac {x - e^{3}}{x}\right ) + 8\right ) + 1} \]
integrate((((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4*log(( x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/x)+(-4 *x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2+6*x)* exp(3)-x^4-2*x^3-6*x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+12*x+5)*exp(3)-2*x^ 4-4*x^3-12*x^2)/(((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4 *log((x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/ x)+(-4*x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2 +x)*exp(3)-x^4-2*x^3-x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+2*x)*exp(3)-2*x^4 -4*x^3-2*x^2),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=x + \frac {5}{- x + \log {\left (4 \log {\left (\frac {x - e^{3}}{x} \right )} + 8 \right )} - 1} \]
integrate((((x*exp(3)-x**2)*ln((x-exp(3))/x)+2*x*exp(3)-2*x**2)*ln(4*ln((x -exp(3))/x)+8)**2+(((-2*x**2-2*x)*exp(3)+2*x**3+2*x**2)*ln((x-exp(3))/x)+( -4*x**2-4*x)*exp(3)+4*x**3+4*x**2)*ln(4*ln((x-exp(3))/x)+8)+((x**3+2*x**2+ 6*x)*exp(3)-x**4-2*x**3-6*x**2)*ln((x-exp(3))/x)+(2*x**3+4*x**2+12*x+5)*ex p(3)-2*x**4-4*x**3-12*x**2)/(((x*exp(3)-x**2)*ln((x-exp(3))/x)+2*x*exp(3)- 2*x**2)*ln(4*ln((x-exp(3))/x)+8)**2+(((-2*x**2-2*x)*exp(3)+2*x**3+2*x**2)* ln((x-exp(3))/x)+(-4*x**2-4*x)*exp(3)+4*x**3+4*x**2)*ln(4*ln((x-exp(3))/x) +8)+((x**3+2*x**2+x)*exp(3)-x**4-2*x**3-x**2)*ln((x-exp(3))/x)+(2*x**3+4*x **2+2*x)*exp(3)-2*x**4-4*x**3-2*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).
Time = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=\frac {x^{2} - x {\left (2 \, \log \left (2\right ) - 1\right )} - x \log \left (\log \left (x - e^{3}\right ) - \log \left (x\right ) + 2\right ) - 5}{x - 2 \, \log \left (2\right ) - \log \left (\log \left (x - e^{3}\right ) - \log \left (x\right ) + 2\right ) + 1} \]
integrate((((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4*log(( x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/x)+(-4 *x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2+6*x)* exp(3)-x^4-2*x^3-6*x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+12*x+5)*exp(3)-2*x^ 4-4*x^3-12*x^2)/(((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4 *log((x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/ x)+(-4*x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2 +x)*exp(3)-x^4-2*x^3-x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+2*x)*exp(3)-2*x^4 -4*x^3-2*x^2),x, algorithm=\
(x^2 - x*(2*log(2) - 1) - x*log(log(x - e^3) - log(x) + 2) - 5)/(x - 2*log (2) - log(log(x - e^3) - log(x) + 2) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1737 vs. \(2 (28) = 56\).
Time = 2.30 (sec) , antiderivative size = 1737, normalized size of antiderivative = 59.90 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=\text {Too large to display} \]
integrate((((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4*log(( x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/x)+(-4 *x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2+6*x)* exp(3)-x^4-2*x^3-6*x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+12*x+5)*exp(3)-2*x^ 4-4*x^3-12*x^2)/(((x*exp(3)-x^2)*log((x-exp(3))/x)+2*x*exp(3)-2*x^2)*log(4 *log((x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*log((x-exp(3))/ x)+(-4*x^2-4*x)*exp(3)+4*x^3+4*x^2)*log(4*log((x-exp(3))/x)+8)+((x^3+2*x^2 +x)*exp(3)-x^4-2*x^3-x^2)*log((x-exp(3))/x)+(2*x^3+4*x^2+2*x)*exp(3)-2*x^4 -4*x^3-2*x^2),x, algorithm=\
(x^4*log(x - e^3)*log((x - e^3)/x) - x^3*e^3*log(x - e^3)*log((x - e^3)/x) - x^4*log(x)*log((x - e^3)/x) + x^3*e^3*log(x)*log((x - e^3)/x) - x^3*log (x - e^3)*log((x - e^3)/x)*log(4*log((x - e^3)/x) + 8) + x^2*e^3*log(x - e ^3)*log((x - e^3)/x)*log(4*log((x - e^3)/x) + 8) + x^3*log(x)*log((x - e^3 )/x)*log(4*log((x - e^3)/x) + 8) - x^2*e^3*log(x)*log((x - e^3)/x)*log(4*l og((x - e^3)/x) + 8) + 2*x^4*log(x - e^3) - 2*x^3*e^3*log(x - e^3) - 2*x^4 *log(x) + 2*x^3*e^3*log(x) + 2*x^4*log((x - e^3)/x) - 2*x^3*e^3*log((x - e ^3)/x) + x^3*log(x - e^3)*log((x - e^3)/x) - x^2*e^3*log(x - e^3)*log((x - e^3)/x) - x^3*log(x)*log((x - e^3)/x) + x^2*e^3*log(x)*log((x - e^3)/x) - 2*x^3*log(x - e^3)*log(4*log((x - e^3)/x) + 8) + 2*x^2*e^3*log(x - e^3)*l og(4*log((x - e^3)/x) + 8) + 2*x^3*log(x)*log(4*log((x - e^3)/x) + 8) - 2* x^2*e^3*log(x)*log(4*log((x - e^3)/x) + 8) - 2*x^3*log((x - e^3)/x)*log(4* log((x - e^3)/x) + 8) + 2*x^2*e^3*log((x - e^3)/x)*log(4*log((x - e^3)/x) + 8) + 4*x^4 - 4*x^3*e^3 + 2*x^3*log(x - e^3) - 3*x^2*e^3*log(x - e^3) - 2 *x^3*log(x) + 3*x^2*e^3*log(x) + 2*x^3*log((x - e^3)/x) - 2*x^2*e^3*log((x - e^3)/x) - 5*x^2*log(x - e^3)*log((x - e^3)/x) + 5*x*e^3*log(x - e^3)*lo g((x - e^3)/x) + 5*x^2*log(x)*log((x - e^3)/x) - 5*x*e^3*log(x)*log((x - e ^3)/x) - 4*x^3*log(4*log((x - e^3)/x) + 8) + 4*x^2*e^3*log(4*log((x - e^3) /x) + 8) + x*e^3*log(x - e^3)*log(4*log((x - e^3)/x) + 8) - x*e^3*log(x)*l og(4*log((x - e^3)/x) + 8) + 4*x^3 - 6*x^2*e^3 - 10*x^2*log(x - e^3) + ...
Time = 9.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx=x-\frac {5}{x-\ln \left (4\,\ln \left (\frac {x-{\mathrm {e}}^3}{x}\right )+8\right )+1} \]
int((log((x - exp(3))/x)*(6*x^2 - exp(3)*(6*x + 2*x^2 + x^3) + 2*x^3 + x^4 ) - log(4*log((x - exp(3))/x) + 8)*(log((x - exp(3))/x)*(2*x^2 - exp(3)*(2 *x + 2*x^2) + 2*x^3) - exp(3)*(4*x + 4*x^2) + 4*x^2 + 4*x^3) - exp(3)*(12* x + 4*x^2 + 2*x^3 + 5) - log(4*log((x - exp(3))/x) + 8)^2*(2*x*exp(3) + lo g((x - exp(3))/x)*(x*exp(3) - x^2) - 2*x^2) + 12*x^2 + 4*x^3 + 2*x^4)/(log ((x - exp(3))/x)*(x^2 - exp(3)*(x + 2*x^2 + x^3) + 2*x^3 + x^4) - exp(3)*( 2*x + 4*x^2 + 2*x^3) - log(4*log((x - exp(3))/x) + 8)^2*(2*x*exp(3) + log( (x - exp(3))/x)*(x*exp(3) - x^2) - 2*x^2) - log(4*log((x - exp(3))/x) + 8) *(log((x - exp(3))/x)*(2*x^2 - exp(3)*(2*x + 2*x^2) + 2*x^3) - exp(3)*(4*x + 4*x^2) + 4*x^2 + 4*x^3) + 2*x^2 + 4*x^3 + 2*x^4),x)