\(\int \frac {6+e^{3+x} (-1+x)+(6 x-e^{3+x} x-x^2) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x}))}{(-6 x^2+e^{3+x} x^2+x^3) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x}))+(-6 x+e^{3+x} x+x^2) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x})) \log (\log (\log (\frac {3 x}{-6+e^{3+x}+x})))} \, dx\) [68]

Optimal result

Integrand size = 170, antiderivative size = 23 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=\log \left (\frac {2}{x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )}\right ) \] Output:

ln(2/(x+ln(ln(ln(3*x/(exp(3+x)+x-6))))))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )\right ) \] Input:

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]
 

Output:

-Log[x + Log[Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]]]
 

Rubi [A] (verified)

Time = 2.38 (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

\(\displaystyle -\log \left (x+\log \left (\log \left (\log \left (-\frac {3 x}{-x-e^{x+3}+6}\right )\right )\right )\right )\)

Input:

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]
 

Output:

-Log[x + Log[Log[Log[(-3*x)/(6 - E^(3 + x) - x)]]]]
 

Defintions of rubi rules used

rule 7235

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L 
og[RemoveContent[y, x]], x] /;  !FalseQ[q]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

Maple [A] (verified)

Time = 175.71 (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	\(-\ln \left (x +\ln \left (\ln \left (\ln \left (3\right )+\ln \left (x \right )-\ln \left ({\mathrm e}^{3+x}+x -6\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{3+x}+x -6}\right )\right )}{2}\right )\right )\right )\)	\(93\)

Input:

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)
 

Output:

-ln(x+ln(ln(ln(3*x/(exp(3+x)+x-6)))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x + \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right )\right ) \] Input:

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="fricas")
 

Output:

-log(x + log(log(log(3*x/(x + e^(x + 3) - 6)))))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=\text {Timed out} \] Input:

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)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\log \left (x + \log \left (\log \left (\log \left (3\right ) - \log \left (x + e^{\left (x + 3\right )} - 6\right ) + \log \left (x\right )\right )\right )\right ) \] Input:

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="maxima")
 

Output:

-log(x + log(log(log(3) - log(x + e^(x + 3) - 6) + log(x))))
 

Giac [F]

\[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=\int { -\frac {{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) - {\left (x - 1\right )} e^{\left (x + 3\right )} - 6}{{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right ) + {\left (x^{3} + x^{2} e^{\left (x + 3\right )} - 6 \, x^{2}\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )} \,d x } \] Input:

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="giac")
 

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 4.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\ln \left (x+\ln \left (\ln \left (\ln \left (\frac {3\,x}{x+{\mathrm {e}}^{x+3}-6}\right )\right )\right )\right ) \] Input:

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 
)
 

Output:

-log(x + log(log(log((3*x)/(x + exp(x + 3) - 6)))))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {6+e^{3+x} (-1+x)+\left (6 x-e^{3+x} x-x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )}{\left (-6 x^2+e^{3+x} x^2+x^3\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )+\left (-6 x+e^{3+x} x+x^2\right ) \log \left (\frac {3 x}{-6+e^{3+x}+x}\right ) \log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right ) \log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {3 x}{e^{x} e^{3}+x -6}\right )\right )\right )+x \right ) \] Input:

int(((-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)
 

Output:

 - log(log(log(log((3*x)/(e**x*e**3 + x - 6)))) + x)