Integrand size = 170, antiderivative size = 23 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=\log \left (\frac {2}{x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )}\right ) \]
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )\right ) \]
Integrate[(6 + E^(3 + x)*(-1 + x) + (6*x - E^(3 + x)*x - x^2)*Log[(3*x)/(- 6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]])/((-6*x^2 + E^(3 + x)*x^2 + x^3)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]] + (-6*x + E^(3 + x)*x + x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Lo g[Log[(3*x)/(-6 + E^(3 + x) + x)]]*Log[Log[Log[(3*x)/(-6 + E^(3 + x) + x)] ]]),x]
Time = 1.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7292, 7235}
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.
| \(\displaystyle \int \frac {\left (-x^2-e^{x+3} x+6 x\right ) \log \left (\frac {3 x}{x+e^{x+3}-6}\right ) \log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )+e^{x+3} (x-1)+6}{\left (x^2+e^{x+3} x-6 x\right ) \log \left (\frac {3 x}{x+e^{x+3}-6}\right ) \log \left (\log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )\right ) \log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )+\left (x^3+e^{x+3} x^2-6 x^2\right ) \log \left (\frac {3 x}{x+e^{x+3}-6}\right ) \log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (-x^2-e^{x+3} x+6 x\right ) \log \left (\frac {3 x}{x+e^{x+3}-6}\right ) \log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )-e^{x+3} (x-1)-6}{\left (-x-e^{x+3}+6\right ) x \log \left (\frac {3 x}{x+e^{x+3}-6}\right ) \log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right ) \left (x+\log \left (\log \left (\log \left (\frac {3 x}{x+e^{x+3}-6}\right )\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 7235 |
| \(\displaystyle -\log \left (x+\log \left (\log \left (\log \left (-\frac {3 x}{-x-e^{x+3}+6}\right )\right )\right )\right )\) |
Int[(6 + E^(3 + x)*(-1 + x) + (6*x - E^(3 + x)*x - x^2)*Log[(3*x)/(-6 + E^ (3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]])/((-6*x^2 + E^(3 + x)*x ^2 + x^3)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]] + (-6*x + E^(3 + x)*x + x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[ (3*x)/(-6 + E^(3 + x) + x)]]*Log[Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]]),x]
3.1.68.3.1 Defintions of rubi rules used
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 187.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
| method | result | size |
| parallelrisch | \(-\ln \left (x +\ln \left (\ln \left (\ln \left (\frac {3 x}{{\mathrm e}^{3+x}+x -6}\right )\right )\right )\right )\) | \(21\) |
| risch | \(-\ln \left (x +\ln \left (\ln \left (\ln \left (3\right )+\ln \left (x \right )-\ln \left ({\mathrm e}^{3+x}+x -6\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3+x}+x -6}\right )\right )}{2}\right )\right )\right )\) | \(93\) |
int(((-exp(3+x)*x-x^2+6*x)*ln(3*x/(exp(3+x)+x-6))*ln(ln(3*x/(exp(3+x)+x-6) ))+(-1+x)*exp(3+x)+6)/((exp(3+x)*x+x^2-6*x)*ln(3*x/(exp(3+x)+x-6))*ln(ln(3 *x/(exp(3+x)+x-6)))*ln(ln(ln(3*x/(exp(3+x)+x-6))))+(x^2*exp(3+x)+x^3-6*x^2 )*ln(3*x/(exp(3+x)+x-6))*ln(ln(3*x/(exp(3+x)+x-6)))),x,method=_RETURNVERBO SE)
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x + \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right )\right ) \]
integrate(((-exp(3+x)*x-x^2+6*x)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp( 3+x)+x-6)))+(-1+x)*exp(3+x)+6)/((exp(3+x)*x+x^2-6*x)*log(3*x/(exp(3+x)+x-6 ))*log(log(3*x/(exp(3+x)+x-6)))*log(log(log(3*x/(exp(3+x)+x-6))))+(x^2*exp (3+x)+x^3-6*x^2)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))),x, algorithm=\
Time = 143.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=- \log {\left (x + \log {\left (\log {\left (\log {\left (\frac {3 x}{x + e^{x + 3} - 6} \right )} \right )} \right )} \right )} \]
integrate(((-exp(3+x)*x-x**2+6*x)*ln(3*x/(exp(3+x)+x-6))*ln(ln(3*x/(exp(3+ x)+x-6)))+(-1+x)*exp(3+x)+6)/((exp(3+x)*x+x**2-6*x)*ln(3*x/(exp(3+x)+x-6)) *ln(ln(3*x/(exp(3+x)+x-6)))*ln(ln(ln(3*x/(exp(3+x)+x-6))))+(x**2*exp(3+x)+ x**3-6*x**2)*ln(3*x/(exp(3+x)+x-6))*ln(ln(3*x/(exp(3+x)+x-6)))),x)
Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x + \log \left (\log \left (\log \left (3\right ) - \log \left (x + e^{\left (x + 3\right )} - 6\right ) + \log \left (x\right )\right )\right )\right ) \]
integrate(((-exp(3+x)*x-x^2+6*x)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp( 3+x)+x-6)))+(-1+x)*exp(3+x)+6)/((exp(3+x)*x+x^2-6*x)*log(3*x/(exp(3+x)+x-6 ))*log(log(3*x/(exp(3+x)+x-6)))*log(log(log(3*x/(exp(3+x)+x-6))))+(x^2*exp (3+x)+x^3-6*x^2)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))),x, algorithm=\
\[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=\int { -\frac {{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) - {\left (x - 1\right )} e^{\left (x + 3\right )} - 6}{{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right ) + {\left (x^{3} + x^{2} e^{\left (x + 3\right )} - 6 \, x^{2}\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )} \,d x } \]
integrate(((-exp(3+x)*x-x^2+6*x)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp( 3+x)+x-6)))+(-1+x)*exp(3+x)+6)/((exp(3+x)*x+x^2-6*x)*log(3*x/(exp(3+x)+x-6 ))*log(log(3*x/(exp(3+x)+x-6)))*log(log(log(3*x/(exp(3+x)+x-6))))+(x^2*exp (3+x)+x^3-6*x^2)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))),x, algorithm=\
integrate(-((x^2 + x*e^(x + 3) - 6*x)*log(3*x/(x + e^(x + 3) - 6))*log(log (3*x/(x + e^(x + 3) - 6))) - (x - 1)*e^(x + 3) - 6)/((x^2 + x*e^(x + 3) - 6*x)*log(3*x/(x + e^(x + 3) - 6))*log(log(3*x/(x + e^(x + 3) - 6)))*log(lo g(log(3*x/(x + e^(x + 3) - 6)))) + (x^3 + x^2*e^(x + 3) - 6*x^2)*log(3*x/( x + e^(x + 3) - 6))*log(log(3*x/(x + e^(x + 3) - 6)))), x)
Time = 11.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\ln \left (x+\ln \left (\ln \left (\ln \left (\frac {3\,x}{x+{\mathrm {e}}^{x+3}-6}\right )\right )\right )\right ) \]
int((exp(x + 3)*(x - 1) - log(log((3*x)/(x + exp(x + 3) - 6)))*log((3*x)/( x + exp(x + 3) - 6))*(x*exp(x + 3) - 6*x + x^2) + 6)/(log(log((3*x)/(x + e xp(x + 3) - 6)))*log((3*x)/(x + exp(x + 3) - 6))*(x^2*exp(x + 3) - 6*x^2 + x^3) + log(log((3*x)/(x + exp(x + 3) - 6)))*log(log(log((3*x)/(x + exp(x + 3) - 6))))*log((3*x)/(x + exp(x + 3) - 6))*(x*exp(x + 3) - 6*x + x^2)),x )