\(\int \frac {(6 x+2 x^2) \log (\frac {3+x}{3})+((2 x-x^2) \log (2-x)+(6-13 x+x^2+2 x^3) \log (2-x) \log (\frac {3+x}{3})) \log (\log (2-x))}{(-6 x+x^2+x^3) \log (2-x) \log (\frac {3+x}{3}) \log (\log (2-x))} \, dx\) [5068]

Optimal result

Integrand size = 112, antiderivative size = 26 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x-\log \left (\frac {x \log \left (\frac {3+x}{3}\right )}{\log ^2(\log (2-x))}\right ) \]

[Out]

2*x-ln(ln(1+1/3*x)/ln(ln(2-x))^2*x)

Rubi [F]

\[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=\int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx \]

[In]

Int[((6*x + 2*x^2)*Log[(3 + x)/3] + ((2*x - x^2)*Log[2 - x] + (6 - 13*x + x^2 + 2*x^3)*Log[2 - x]*Log[(3 + x)/
3])*Log[Log[2 - x]])/((-6*x + x^2 + x^3)*Log[2 - x]*Log[(3 + x)/3]*Log[Log[2 - x]]),x]

[Out]

2*x - ((3 + Log[27])*Log[Log[(3 + x)/3]])/3 - Log[(3 + x)/3]*Log[Log[(3 + x)/3]] + Log[3 + x]*Log[Log[(3 + x)/
3]] + 2*Log[Log[Log[2 - x]]] + (Log[27]*Defer[Int][1/(x*Log[1 + x/3]), x])/3 - Defer[Int][Log[3 + x]/(x*Log[1
+ x/3]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{x \left (-6+x+x^2\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx \\ & = \int \left (\frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )-3 \log (3+x)}{x (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {2}{(-2+x) \log (2-x) \log (\log (2-x))}\right ) \, dx \\ & = 2 \int \frac {1}{(-2+x) \log (2-x) \log (\log (2-x))} \, dx+\int \frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )-3 \log (3+x)}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 \log (\log (\log (2-x)))+\int \left (\frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {3 \log (3+x)}{(-3-x) x \log \left (1+\frac {x}{3}\right )}\right ) \, dx \\ & = 2 \log (\log (\log (2-x)))+3 \int \frac {\log (3+x)}{(-3-x) x \log \left (1+\frac {x}{3}\right )} \, dx+\int \frac {-x+\log (27)+5 x \log \left (\frac {3+x}{3}\right )+2 x^2 \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 \log (\log (\log (2-x)))+3 \int \left (-\frac {\log (3+x)}{3 x \log \left (1+\frac {x}{3}\right )}+\frac {\log (3+x)}{3 (3+x) \log \left (1+\frac {x}{3}\right )}\right ) \, dx+\int \frac {-x+\log (27)+x (5+2 x) \log \left (\frac {3+x}{3}\right )}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 \log (\log (\log (2-x)))+\int \left (\frac {5+2 x}{3+x}+\frac {-x+\log (27)}{x (3+x) \log \left (1+\frac {x}{3}\right )}\right ) \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx+\int \frac {\log (3+x)}{(3+x) \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 \log (\log (\log (2-x)))+3 \text {Subst}\left (\int \frac {\log (3 x)}{3 x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\int \frac {5+2 x}{3+x} \, dx+\int \frac {-x+\log (27)}{x (3+x) \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 \log (\log (\log (2-x)))+\int \left (2+\frac {1}{-3-x}\right ) \, dx+\int \left (\frac {-3-\log (27)}{3 (3+x) \log \left (1+\frac {x}{3}\right )}+\frac {\log (27)}{3 x \log \left (1+\frac {x}{3}\right )}\right ) \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx+\text {Subst}\left (\int \frac {\log (3 x)}{x \log (x)} \, dx,x,1+\frac {x}{3}\right ) \\ & = 2 x-\log (3+x)+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \int \frac {1}{(3+x) \log \left (1+\frac {x}{3}\right )} \, dx+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx-\text {Subst}\left (\int \frac {\log (\log (x))}{x} \, dx,x,1+\frac {x}{3}\right ) \\ & = 2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+(-3-\log (27)) \text {Subst}\left (\int \frac {1}{3 x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,1+\frac {x}{3}\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 x-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} (-3-\log (27)) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {3+x}{3}\right )\right )+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx \\ & = 2 x-\frac {1}{3} (3+\log (27)) \log \left (\log \left (\frac {3+x}{3}\right )\right )-\log \left (\frac {3+x}{3}\right ) \log \left (\log \left (\frac {3+x}{3}\right )\right )+\log (3+x) \log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x)))+\frac {1}{3} \log (27) \int \frac {1}{x \log \left (1+\frac {x}{3}\right )} \, dx-\int \frac {\log (3+x)}{x \log \left (1+\frac {x}{3}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x-\log (x)-\log \left (\log \left (\frac {3+x}{3}\right )\right )+2 \log (\log (\log (2-x))) \]

[In]

Integrate[((6*x + 2*x^2)*Log[(3 + x)/3] + ((2*x - x^2)*Log[2 - x] + (6 - 13*x + x^2 + 2*x^3)*Log[2 - x]*Log[(3
+ x)/3])*Log[Log[2 - x]])/((-6*x + x^2 + x^3)*Log[2 - x]*Log[(3 + x)/3]*Log[Log[2 - x]]),x]

[Out]

2*x - Log[x] - Log[Log[(3 + x)/3]] + 2*Log[Log[Log[2 - x]]]

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08

method	result	size
risch	\(2 x -\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\)	\(28\)
parts	\(2 x -\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\)	\(28\)
parallelrisch	\(11-\ln \left (x \right )-\ln \left (\ln \left (1+\frac {x}{3}\right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )+2 x\)	\(29\)
default	\(2 x +6-\ln \left (x \right )-\ln \left (\ln \left (3\right )-\ln \left (3+x \right )\right )+2 \ln \left (\ln \left (\ln \left (2-x \right )\right )\right )\)	\(32\)

[In]

int((((2*x^3+x^2-13*x+6)*ln(2-x)*ln(1+1/3*x)+(-x^2+2*x)*ln(2-x))*ln(ln(2-x))+(2*x^2+6*x)*ln(1+1/3*x))/(x^3+x^2
-6*x)/ln(2-x)/ln(1+1/3*x)/ln(ln(2-x)),x,method=_RETURNVERBOSE)

[Out]

2*x-ln(x)-ln(ln(1+1/3*x))+2*ln(ln(ln(2-x)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (x\right ) - \log \left (\log \left (\frac {1}{3} \, x + 1\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \]

[In]

integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))*log(log(2-x))+(2*x^2+6*x)*log(1+1/3*
x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x)/log(log(2-x)),x, algorithm="fricas")

[Out]

2*x - log(x) - log(log(1/3*x + 1)) + 2*log(log(log(-x + 2)))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 x - \log {\left (x \right )} - \log {\left (\log {\left (\frac {x}{3} + 1 \right )} \right )} + 2 \log {\left (\log {\left (\log {\left (2 - x \right )} \right )} \right )} \]

[In]

integrate((((2*x**3+x**2-13*x+6)*ln(2-x)*ln(1+1/3*x)+(-x**2+2*x)*ln(2-x))*ln(ln(2-x))+(2*x**2+6*x)*ln(1+1/3*x)
)/(x**3+x**2-6*x)/ln(2-x)/ln(1+1/3*x)/ln(ln(2-x)),x)

[Out]

2*x - log(x) - log(log(x/3 + 1)) + 2*log(log(log(2 - x)))

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (x\right ) - \log \left (-\log \left (3\right ) + \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) \]

[In]

integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))*log(log(2-x))+(2*x^2+6*x)*log(1+1/3*
x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x)/log(log(2-x)),x, algorithm="maxima")

[Out]

2*x - log(x) - log(-log(3) + log(x + 3)) + 2*log(log(log(-x + 2)))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2 \, x - \log \left (-x\right ) - \log \left (-\log \left (3\right ) + \log \left (x + 3\right )\right ) + 2 \, \log \left (\log \left (\log \left (-x + 2\right )\right )\right ) - 4 \]

[In]

integrate((((2*x^3+x^2-13*x+6)*log(2-x)*log(1+1/3*x)+(-x^2+2*x)*log(2-x))*log(log(2-x))+(2*x^2+6*x)*log(1+1/3*
x))/(x^3+x^2-6*x)/log(2-x)/log(1+1/3*x)/log(log(2-x)),x, algorithm="giac")

[Out]

2*x - log(-x) - log(-log(3) + log(x + 3)) + 2*log(log(log(-x + 2))) - 4

Mupad [B] (verification not implemented)

Time = 11.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{3}\right )+\left (\left (2 x-x^2\right ) \log (2-x)+\left (6-13 x+x^2+2 x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right )\right ) \log (\log (2-x))}{\left (-6 x+x^2+x^3\right ) \log (2-x) \log \left (\frac {3+x}{3}\right ) \log (\log (2-x))} \, dx=2\,x-\ln \left (\ln \left (\frac {x}{3}+1\right )\right )+2\,\ln \left (\ln \left (\ln \left (2-x\right )\right )\right )-\ln \left (x\right ) \]

[In]

int((log(log(2 - x))*(log(2 - x)*(2*x - x^2) + log(2 - x)*log(x/3 + 1)*(x^2 - 13*x + 2*x^3 + 6)) + log(x/3 + 1
)*(6*x + 2*x^2))/(log(log(2 - x))*log(2 - x)*log(x/3 + 1)*(x^2 - 6*x + x^3)),x)

[Out]

2*x - log(log(x/3 + 1)) + 2*log(log(log(2 - x))) - log(x)