\(\int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} (-648 x-648 x^2-192 x^3-16 x^4) \log (\frac {27+18 x+2 x^2}{9+6 x+x^2})}{(81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7) \log (\frac {27+18 x+2 x^2}{9+6 x+x^2})} \, dx\) [2067]

Optimal result

Integrand size = 134, antiderivative size = 31 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^4}{1+x^2}}+\log \left (2 \log \left (3-\frac {x^2}{(3+x)^2}\right )\right ) \] Output:

ln(2*ln(3-x^2/(3+x)^2))+exp(4*exp(4)/(x^2+1))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^4}{1+x^2}}+\log \left (\log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )\right ) \] Input:

Integrate[(-6*x - 12*x^3 - 6*x^5 + E^(4 + (4*E^4)/(1 + x^2))*(-648*x - 648 
*x^2 - 192*x^3 - 16*x^4)*Log[(27 + 18*x + 2*x^2)/(9 + 6*x + x^2)])/((81 + 
81*x + 186*x^2 + 164*x^3 + 129*x^4 + 85*x^5 + 24*x^6 + 2*x^7)*Log[(27 + 18 
*x + 2*x^2)/(9 + 6*x + x^2)]),x]
 

Output:

E^((4*E^4)/(1 + x^2)) + Log[Log[(27 + 18*x + 2*x^2)/(3 + x)^2]]
 

Rubi [F]

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.

\(\displaystyle \int \frac {-6 x^5-12 x^3+e^{\frac {4 e^4}{x^2+1}+4} \left (-16 x^4-192 x^3-648 x^2-648 x\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )-6 x}{\left (2 x^7+24 x^6+85 x^5+129 x^4+164 x^3+186 x^2+81 x+81\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {-6 x^5-12 x^3+e^{\frac {4 e^4}{x^2+1}+4} \left (-16 x^4-192 x^3-648 x^2-648 x\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )-6 x}{900 (x+3) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )}+\frac {(409533-270911 x) \left (-6 x^5-12 x^3+e^{\frac {4 e^4}{x^2+1}+4} \left (-16 x^4-192 x^3-648 x^2-648 x\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )-6 x\right )}{90060100 \left (x^2+1\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )}+\frac {8 (8347 x+59154) \left (-6 x^5-12 x^3+e^{\frac {4 e^4}{x^2+1}+4} \left (-16 x^4-192 x^3-648 x^2-648 x\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )-6 x\right )}{8105409 \left (2 x^2+18 x+27\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )}+\frac {(57-79 x) \left (-6 x^5-12 x^3+e^{\frac {4 e^4}{x^2+1}+4} \left (-16 x^4-192 x^3-648 x^2-648 x\right ) \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )-6 x\right )}{9490 \left (x^2+1\right )^2 \log \left (\frac {2 x^2+18 x+27}{x^2+6 x+9}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -4 \sqrt {3} \int \frac {1}{\left (-4 x+6 \sqrt {3}-18\right ) \log \left (\frac {2 x^2+18 x+27}{(x+3)^2}\right )}dx-2 \int \frac {1}{(x+3) \log \left (\frac {2 x^2+18 x+27}{(x+3)^2}\right )}dx+4 \left (1-\sqrt {3}\right ) \int \frac {1}{\left (4 x-6 \sqrt {3}+18\right ) \log \left (\frac {2 x^2+18 x+27}{(x+3)^2}\right )}dx+4 \left (1+\sqrt {3}\right ) \int \frac {1}{\left (4 x+6 \sqrt {3}+18\right ) \log \left (\frac {2 x^2+18 x+27}{(x+3)^2}\right )}dx-4 \sqrt {3} \int \frac {1}{\left (4 x+6 \sqrt {3}+18\right ) \log \left (\frac {2 x^2+18 x+27}{(x+3)^2}\right )}dx+e^{\frac {4 e^4}{x^2+1}}\)

Input:

Int[(-6*x - 12*x^3 - 6*x^5 + E^(4 + (4*E^4)/(1 + x^2))*(-648*x - 648*x^2 - 
 192*x^3 - 16*x^4)*Log[(27 + 18*x + 2*x^2)/(9 + 6*x + x^2)])/((81 + 81*x + 
 186*x^2 + 164*x^3 + 129*x^4 + 85*x^5 + 24*x^6 + 2*x^7)*Log[(27 + 18*x + 2 
*x^2)/(9 + 6*x + x^2)]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 28.77 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03

method	result	size
default	\(\ln \left (\ln \left (2-\frac {9}{\left (3+x \right )^{2}}+\frac {6}{3+x}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(32\)
parts	\(\ln \left (\ln \left (2-\frac {9}{\left (3+x \right )^{2}}+\frac {6}{3+x}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(32\)
parallelrisch	\(\ln \left (\ln \left (\frac {2 x^{2}+18 x +27}{x^{2}+6 x +9}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\)	\(37\)
risch	\({\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}+\ln \left (\ln \left (x^{2}+9 x +\frac {27}{2}\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{2}-\pi \operatorname {csgn}\left (i \left (3+x \right )\right )^{2} \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \left (3+x \right )\right ) \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-\pi \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )}^{3}+2 i \ln \left (2\right )-4 i \ln \left (3+x \right )\right )}{2}\right )\)	\(223\)

Input:

int(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*ln((2*x^ 
2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+164 
*x^3+186*x^2+81*x+81)/ln((2*x^2+18*x+27)/(x^2+6*x+9)),x,method=_RETURNVERB 
OSE)
 

Output:

ln(ln(2-9/(3+x)^2+6/(3+x)))+exp(4*exp(4)/(x^2+1))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx={\left (e^{4} \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) + e^{\left (\frac {4 \, {\left (x^{2} + e^{4} + 1\right )}}{x^{2} + 1}\right )}\right )} e^{\left (-4\right )} \] Input:

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*lo 
g((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129* 
x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorithm 
="fricas")
 

Output:

(e^4*log(log((2*x^2 + 18*x + 27)/(x^2 + 6*x + 9))) + e^(4*(x^2 + e^4 + 1)/ 
(x^2 + 1)))*e^(-4)
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^{4}}{x^{2} + 1}} + \log {\left (\log {\left (\frac {2 x^{2} + 18 x + 27}{x^{2} + 6 x + 9} \right )} \right )} \] Input:

integrate(((-16*x**4-192*x**3-648*x**2-648*x)*exp(4)*exp(4*exp(4)/(x**2+1) 
)*ln((2*x**2+18*x+27)/(x**2+6*x+9))-6*x**5-12*x**3-6*x)/(2*x**7+24*x**6+85 
*x**5+129*x**4+164*x**3+186*x**2+81*x+81)/ln((2*x**2+18*x+27)/(x**2+6*x+9) 
),x)
 

Output:

exp(4*exp(4)/(x**2 + 1)) + log(log((2*x**2 + 18*x + 27)/(x**2 + 6*x + 9)))
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\left (\frac {4 \, e^{4}}{x^{2} + 1}\right )} + \log \left (\log \left (2 \, x^{2} + 18 \, x + 27\right ) - 2 \, \log \left (x + 3\right )\right ) \] Input:

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*lo 
g((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129* 
x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorithm 
="maxima")
 

Output:

e^(4*e^4/(x^2 + 1)) + log(log(2*x^2 + 18*x + 27) - 2*log(x + 3))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (29) = 58\).

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\left (\frac {4 \, x^{2}}{x^{2} + 1} + \frac {4 \, e^{4}}{x^{2} + 1} + \frac {4}{x^{2} + 1} - 4\right )} + \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) \] Input:

integrate(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*lo 
g((2*x^2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129* 
x^4+164*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x, algorithm 
="giac")
 

Output:

e^(4*x^2/(x^2 + 1) + 4*e^4/(x^2 + 1) + 4/(x^2 + 1) - 4) + log(log((2*x^2 + 
 18*x + 27)/(x^2 + 6*x + 9)))
 

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=\ln \left (\ln \left (\frac {2\,x^2+18\,x+27}{x^2+6\,x+9}\right )\right )+{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^4}{x^2+1}} \] Input:

int(-(6*x + 12*x^3 + 6*x^5 + exp(4)*exp((4*exp(4))/(x^2 + 1))*log((18*x + 
2*x^2 + 27)/(6*x + x^2 + 9))*(648*x + 648*x^2 + 192*x^3 + 16*x^4))/(log((1 
8*x + 2*x^2 + 27)/(6*x + x^2 + 9))*(81*x + 186*x^2 + 164*x^3 + 129*x^4 + 8 
5*x^5 + 24*x^6 + 2*x^7 + 81)),x)
 

Output:

log(log((18*x + 2*x^2 + 27)/(6*x + x^2 + 9))) + exp((4*exp(4))/(x^2 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )} \, dx=e^{\frac {4 e^{4}}{x^{2}+1}}+\mathrm {log}\left (\mathrm {log}\left (\frac {2 x^{2}+18 x +27}{x^{2}+6 x +9}\right )\right ) \] Input:

int(((-16*x^4-192*x^3-648*x^2-648*x)*exp(4)*exp(4*exp(4)/(x^2+1))*log((2*x 
^2+18*x+27)/(x^2+6*x+9))-6*x^5-12*x^3-6*x)/(2*x^7+24*x^6+85*x^5+129*x^4+16 
4*x^3+186*x^2+81*x+81)/log((2*x^2+18*x+27)/(x^2+6*x+9)),x)
 

Output:

e**((4*e**4)/(x**2 + 1)) + log(log((2*x**2 + 18*x + 27)/(x**2 + 6*x + 9)))