\(\int \frac {16 x-2 x^4+(8 x+2 x^4) \log (\frac {4+x^3}{x^2}) \log (\frac {3}{11} \log (\frac {4+x^3}{x^2}))+(-4-x^3) \log (\frac {4+x^3}{x^2}) \log ^3(\frac {3}{11} \log (\frac {4+x^3}{x^2}))}{(4+x^3) \log (\frac {4+x^3}{x^2}) \log ^3(\frac {3}{11} \log (\frac {4+x^3}{x^2}))} \, dx\) [2587]

Optimal result

Integrand size = 116, antiderivative size = 23 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=-x+\frac {x^2}{\log ^2\left (\frac {3}{11} \log \left (\frac {4}{x^2}+x\right )\right )} \] Output:

1/ln(3/11*ln(x+4/x^2))^2*x^2-x
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=-x+\frac {x^2}{\log ^2\left (\frac {3}{11} \log \left (\frac {4}{x^2}+x\right )\right )} \] Input:

Integrate[(16*x - 2*x^4 + (8*x + 2*x^4)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + 
 x^3)/x^2])/11] + (-4 - x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2]) 
/11]^3)/((4 + x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3),x]
 

Output:

-x + x^2/Log[(3*Log[4/x^2 + x])/11]^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.

\(\displaystyle \int \frac {-2 x^4+\left (-x^3-4\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )+\left (2 x^4+8 x\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )+16 x}{\left (x^3+4\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (-\frac {2 \left (x^3-8\right ) x}{\left (x^3+4\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}+\frac {2 x}{\log ^2\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {x}{\log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}dx-4 \sqrt [3]{2} \int \frac {1}{\left (x+2^{2/3}\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}dx-4 (-1)^{2/3} \sqrt [3]{2} \int \frac {1}{\left (2^{2/3}-\sqrt [3]{-1} x\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}dx+4 \sqrt [3]{-2} \int \frac {1}{\left ((-1)^{2/3} x+2^{2/3}\right ) \log \left (\frac {x^3+4}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}dx+2 \int \frac {x}{\log ^2\left (\frac {3}{11} \log \left (\frac {x^3+4}{x^2}\right )\right )}dx-x\)

Input:

Int[(16*x - 2*x^4 + (8*x + 2*x^4)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/ 
x^2])/11] + (-4 - x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3 
)/((4 + x^3)*Log[(4 + x^3)/x^2]*Log[(3*Log[(4 + x^3)/x^2])/11]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83

method	result	size
parallelrisch	\(\frac {-8 {\ln \left (\frac {3 \ln \left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )}^{2} x +8 x^{2}}{8 {\ln \left (\frac {3 \ln \left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )}^{2}}\)	\(42\)

Input:

int(((-x^3-4)*ln((x^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))^3+(2*x^4+8*x)*ln((x 
^3+4)/x^2)*ln(3/11*ln((x^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/ln((x^3+4)/x^2)/ln 
(3/11*ln((x^3+4)/x^2))^3,x,method=_RETURNVERBOSE)
 

Output:

1/8*(-8*ln(3/11*ln((x^3+4)/x^2))^2*x+8*x^2)/ln(3/11*ln((x^3+4)/x^2))^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=-\frac {x \log \left (\frac {3}{11} \, \log \left (\frac {x^{3} + 4}{x^{2}}\right )\right )^{2} - x^{2}}{\log \left (\frac {3}{11} \, \log \left (\frac {x^{3} + 4}{x^{2}}\right )\right )^{2}} \] Input:

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8 
*x)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x 
^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="fricas")
 

Output:

-(x*log(3/11*log((x^3 + 4)/x^2))^2 - x^2)/log(3/11*log((x^3 + 4)/x^2))^2
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {3 \log {\left (\frac {x^{3} + 4}{x^{2}} \right )}}{11} \right )}^{2}} - x \] Input:

integrate(((-x**3-4)*ln((x**3+4)/x**2)*ln(3/11*ln((x**3+4)/x**2))**3+(2*x* 
*4+8*x)*ln((x**3+4)/x**2)*ln(3/11*ln((x**3+4)/x**2))-2*x**4+16*x)/(x**3+4) 
/ln((x**3+4)/x**2)/ln(3/11*ln((x**3+4)/x**2))**3,x)
 

Output:

x**2/log(3*log((x**3 + 4)/x**2)/11)**2 - x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (21) = 42\).

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.96 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=\frac {2 \, x {\left (\log \left (11\right ) - \log \left (3\right )\right )} \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \left (x\right )\right ) - x \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \left (x\right )\right )^{2} - {\left (\log \left (11\right )^{2} - 2 \, \log \left (11\right ) \log \left (3\right ) + \log \left (3\right )^{2}\right )} x + x^{2}}{\log \left (11\right )^{2} - 2 \, \log \left (11\right ) \log \left (3\right ) + \log \left (3\right )^{2} - 2 \, {\left (\log \left (11\right ) - \log \left (3\right )\right )} \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \left (x\right )\right ) + \log \left (\log \left (x^{3} + 4\right ) - 2 \, \log \left (x\right )\right )^{2}} \] Input:

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8 
*x)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x 
^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="maxima")
 

Output:

(2*x*(log(11) - log(3))*log(log(x^3 + 4) - 2*log(x)) - x*log(log(x^3 + 4) 
- 2*log(x))^2 - (log(11)^2 - 2*log(11)*log(3) + log(3)^2)*x + x^2)/(log(11 
)^2 - 2*log(11)*log(3) + log(3)^2 - 2*(log(11) - log(3))*log(log(x^3 + 4) 
- 2*log(x)) + log(log(x^3 + 4) - 2*log(x))^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1412 vs. \(2 (21) = 42\).

Time = 2.13 (sec) , antiderivative size = 1412, normalized size of antiderivative = 61.39 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8 
*x)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x 
^3+4)/x^2)/log(3/11*log((x^3+4)/x^2))^3,x, algorithm="giac")
 

Output:

-x + (2*x^5*log(11)*log(x^3 + 4)^2*log((x^3 + 4)/x^2) - 4*x^5*log(11)*log( 
x^3 + 4)*log(x^2)*log((x^3 + 4)/x^2) + 2*x^5*log(11)*log(x^2)^2*log((x^3 + 
 4)/x^2) - 2*x^5*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^ 
2) + 4*x^5*log(x^3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x 
^2) - 2*x^5*log(x^2)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - 2*x^ 
5*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2)^2 + 2*x^5*log(11)*log(x^2)*log(( 
x^3 + 4)/x^2)^2 + 2*x^5*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 
4)/x^2)^2 - 2*x^5*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2)^2 
- x^5*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2) + x^5*log(11)*log(x^2)*log(( 
x^3 + 4)/x^2) + x^5*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x 
^2) - x^5*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) + x^5*log( 
11)*log((x^3 + 4)/x^2)^2 - x^5*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2 
)^2 + x^5*log(x^3 + 4)*log((x^3 + 4)/x^2) - x^5*log(x^2)*log((x^3 + 4)/x^2 
) + 8*x^2*log(11)*log(x^3 + 4)^2*log((x^3 + 4)/x^2) - 16*x^2*log(11)*log(x 
^3 + 4)*log(x^2)*log((x^3 + 4)/x^2) + 8*x^2*log(11)*log(x^2)^2*log((x^3 + 
4)/x^2) - 8*x^2*log(x^3 + 4)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2 
) + 16*x^2*log(x^3 + 4)*log(x^2)*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x 
^2) - 8*x^2*log(x^2)^2*log(3*log((x^3 + 4)/x^2))*log((x^3 + 4)/x^2) - 8*x^ 
2*log(11)*log(x^3 + 4)*log((x^3 + 4)/x^2)^2 + 8*x^2*log(11)*log(x^2)*log(( 
x^3 + 4)/x^2)^2 + 8*x^2*log(x^3 + 4)*log(3*log((x^3 + 4)/x^2))*log((x^3...
 

Mupad [B] (verification not implemented)

Time = 3.31 (sec) , antiderivative size = 254, normalized size of antiderivative = 11.04 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=\frac {x^2}{{\ln \left (\frac {3\,\ln \left (\frac {1}{x^2}\right )}{11}+\frac {3\,\ln \left (x^3+4\right )}{11}\right )}^2}-x+\frac {4\,x^2\,\ln \left (\frac {1}{x^2}\right )}{x^3-8}+\frac {x^5\,\ln \left (\frac {1}{x^2}\right )}{x^3-8}+\frac {4\,x^2\,\ln \left (x^3+4\right )}{x^3-8}+\frac {x^5\,\ln \left (x^3+4\right )}{x^3-8}-\frac {256\,x^2\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}+\frac {12\,x^8\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {x^{11}\,\ln \left (\frac {1}{x^2}\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {256\,x^2\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512}+\frac {12\,x^8\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512}-\frac {x^{11}\,\ln \left (x^3+4\right )}{x^9-24\,x^6+192\,x^3-512} \] Input:

int((16*x - 2*x^4 + log((3*log((x^3 + 4)/x^2))/11)*log((x^3 + 4)/x^2)*(8*x 
 + 2*x^4) - log((3*log((x^3 + 4)/x^2))/11)^3*log((x^3 + 4)/x^2)*(x^3 + 4)) 
/(log((3*log((x^3 + 4)/x^2))/11)^3*log((x^3 + 4)/x^2)*(x^3 + 4)),x)
 

Output:

x^2/log((3*log(1/x^2))/11 + (3*log(x^3 + 4))/11)^2 - x + (4*x^2*log(1/x^2) 
)/(x^3 - 8) + (x^5*log(1/x^2))/(x^3 - 8) + (4*x^2*log(x^3 + 4))/(x^3 - 8) 
+ (x^5*log(x^3 + 4))/(x^3 - 8) - (256*x^2*log(1/x^2))/(192*x^3 - 24*x^6 + 
x^9 - 512) + (12*x^8*log(1/x^2))/(192*x^3 - 24*x^6 + x^9 - 512) - (x^11*lo 
g(1/x^2))/(192*x^3 - 24*x^6 + x^9 - 512) - (256*x^2*log(x^3 + 4))/(192*x^3 
 - 24*x^6 + x^9 - 512) + (12*x^8*log(x^3 + 4))/(192*x^3 - 24*x^6 + x^9 - 5 
12) - (x^11*log(x^3 + 4))/(192*x^3 - 24*x^6 + x^9 - 512)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {16 x-2 x^4+\left (8 x+2 x^4\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log \left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )+\left (-4-x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )}{\left (4+x^3\right ) \log \left (\frac {4+x^3}{x^2}\right ) \log ^3\left (\frac {3}{11} \log \left (\frac {4+x^3}{x^2}\right )\right )} \, dx=\frac {x \left (-{\mathrm {log}\left (\frac {3 \,\mathrm {log}\left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )}^{2}+x \right )}{{\mathrm {log}\left (\frac {3 \,\mathrm {log}\left (\frac {x^{3}+4}{x^{2}}\right )}{11}\right )}^{2}} \] Input:

int(((-x^3-4)*log((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))^3+(2*x^4+8*x)*lo 
g((x^3+4)/x^2)*log(3/11*log((x^3+4)/x^2))-2*x^4+16*x)/(x^3+4)/log((x^3+4)/ 
x^2)/log(3/11*log((x^3+4)/x^2))^3,x)
 

Output:

(x*( - log((3*log((x**3 + 4)/x**2))/11)**2 + x))/log((3*log((x**3 + 4)/x** 
2))/11)**2