\(\int \frac {-96+6 e^4+8 x+(-48 x+3 e^4 x-4 x^2) \log ^2(\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})}{-192 x+12 e^4 x-16 x^2+(192 x^2-12 e^4 x^2+16 x^3) \log (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})+(-48 x^3+3 e^4 x^3-4 x^4) \log ^2(\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2})} \, dx\) [133]

Optimal result

Integrand size = 178, antiderivative size = 29 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=\frac {2}{-2 x+\frac {4}{\log \left (\frac {x}{\left (16-e^4+\frac {4 x}{3}\right )^2}\right )}} \] Output:

2/(4/ln(x/(4/3*x-exp(4)+16)^2)-2*x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=\frac {-1-\frac {2}{-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}}{x} \] Input:

Integrate[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 
 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2)/(-192*x + 12*E^4*x - 16*x 
^2 + (192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 
 24*x) + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 
+ 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2),x]
 

Output:

(-1 - 2/(-2 + x*Log[(9*x)/(3*E^4 - 4*(12 + x))^2]))/x
 

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.

\(\displaystyle \int \frac {\left (-4 x^2+3 e^4 x-48 x\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+8 x+6 e^4-96}{-16 x^2+\left (16 x^3-12 e^4 x^2+192 x^2\right ) \log \left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (-4 x^4+3 e^4 x^3-48 x^3\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+12 e^4 x-192 x} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (-4 x^2+3 e^4 x-48 x\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+8 x+6 e^4-96}{-16 x^2+\left (16 x^3-12 e^4 x^2+192 x^2\right ) \log \left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (-4 x^4+3 e^4 x^3-48 x^3\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (12 e^4-192\right ) x}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-8 x+x \left (4 x-3 e^4+48\right ) \log ^2\left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )+96 \left (1-\frac {e^4}{16}\right )}{x \left (4 x-3 e^4+48\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{x^2}+\frac {2 \left (-4 x^2+\left (56-3 e^4\right ) x+6 \left (16-e^4\right )\right )}{x^2 \left (4 x-3 e^4+48\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )\right )^2}+\frac {4}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {1}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx+4 \int \frac {1}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )}dx+16 \int \frac {1}{\left (-4 x+3 e^4-48\right ) \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx+2 \int \frac {1}{x \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx-\frac {1}{x}\)

Input:

Int[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 9*E^8 
 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2)/(-192*x + 12*E^4*x - 16*x^2 + ( 
192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) 
 + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 + 9*E^ 
8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59

method	result	size
risch	\(-\frac {1}{x}-\frac {2}{x \left (\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2\right )}\)	\(46\)
norman	\(-\frac {\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )}{\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2}\)	\(69\)
parallelrisch	\(-\frac {\ln \left (\frac {9 x}{-24 x \,{\mathrm e}^{4}+16 x^{2}+9 \,{\mathrm e}^{8}-288 \,{\mathrm e}^{4}+384 x +2304}\right )}{\ln \left (\frac {9 x}{-24 x \,{\mathrm e}^{4}+16 x^{2}+9 \,{\mathrm e}^{8}-288 \,{\mathrm e}^{4}+384 x +2304}\right ) x -2}\)	\(71\)

Input:

int(((3*x*exp(4)-4*x^2-48*x)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+ 
384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*ln(9*x/(9*exp 
(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16*x^3+192* 
x^2)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4) 
-16*x^2-192*x),x,method=_RETURNVERBOSE)
 

Output:

-1/x-2/x/(ln(9*x/(9*exp(8)+(-24*x-288)*exp(4)+16*x^2+384*x+2304))*x-2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right ) - 2} \] Input:

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 
16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* 
x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* 
x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 
*x*exp(4)-16*x^2-192*x),x, algorithm="fricas")
 

Output:

-log(9*x/(16*x^2 - 24*(x + 12)*e^4 + 384*x + 9*e^8 + 2304))/(x*log(9*x/(16 
*x^2 - 24*(x + 12)*e^4 + 384*x + 9*e^8 + 2304)) - 2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=- \frac {2}{x^{2} \log {\left (\frac {9 x}{16 x^{2} + 384 x + \left (- 24 x - 288\right ) e^{4} + 2304 + 9 e^{8}} \right )} - 2 x} - \frac {1}{x} \] Input:

integrate(((3*x*exp(4)-4*x**2-48*x)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4) 
+16*x**2+384*x+2304))**2+6*exp(4)+8*x-96)/((3*x**3*exp(4)-4*x**4-48*x**3)* 
ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384*x+2304))**2+(-12*x**2*e 
xp(4)+16*x**3+192*x**2)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384 
*x+2304))+12*x*exp(4)-16*x**2-192*x),x)
 

Output:

-2/(x**2*log(9*x/(16*x**2 + 384*x + (-24*x - 288)*exp(4) + 2304 + 9*exp(8) 
)) - 2*x) - 1/x
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {2 \, \log \left (3\right ) - 2 \, \log \left (4 \, x - 3 \, e^{4} + 48\right ) + \log \left (x\right )}{2 \, x \log \left (3\right ) - 2 \, x \log \left (4 \, x - 3 \, e^{4} + 48\right ) + x \log \left (x\right ) - 2} \] Input:

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 
16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* 
x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* 
x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 
*x*exp(4)-16*x^2-192*x),x, algorithm="maxima")
 

Output:

-(2*log(3) - 2*log(4*x - 3*e^4 + 48) + log(x))/(2*x*log(3) - 2*x*log(4*x - 
 3*e^4 + 48) + x*log(x) - 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).

Time = 1.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right ) - 2} \] Input:

integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 
16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* 
x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* 
x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 
*x*exp(4)-16*x^2-192*x),x, algorithm="giac")
 

Output:

-log(9*x/(16*x^2 - 24*x*e^4 + 384*x + 9*e^8 - 288*e^4 + 2304))/(x*log(9*x/ 
(16*x^2 - 24*x*e^4 + 384*x + 9*e^8 - 288*e^4 + 2304)) - 2)
 

Mupad [B] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {2}{x\,\left (2\,x\,\ln \left (3\right )-x\,\ln \left (384\,x-288\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-24\,x\,{\mathrm {e}}^4+16\,x^2+2304\right )+x\,\ln \left (x\right )-2\right )}-\frac {1}{x} \] Input:

int(-(8*x + 6*exp(4) - log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x 
 + 288) + 2304))^2*(48*x - 3*x*exp(4) + 4*x^2) - 96)/(192*x - 12*x*exp(4) 
- log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304))*(192 
*x^2 - 12*x^2*exp(4) + 16*x^3) + 16*x^2 + log((9*x)/(384*x + 9*exp(8) + 16 
*x^2 - exp(4)*(24*x + 288) + 2304))^2*(48*x^3 - 3*x^3*exp(4) + 4*x^4)),x)
 

Output:

- 2/(x*(2*x*log(3) - x*log(384*x - 288*exp(4) + 9*exp(8) - 24*x*exp(4) + 1 
6*x^2 + 2304) + x*log(x) - 2)) - 1/x
 

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 177, normalized size of antiderivative = 6.10 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=\frac {-2 \,\mathrm {log}\left (3 e^{4}-4 x -48\right ) \mathrm {log}\left (\frac {9 x}{9 e^{8}-24 e^{4} x -288 e^{4}+16 x^{2}+384 x +2304}\right ) x +4 \,\mathrm {log}\left (3 e^{4}-4 x -48\right )-\mathrm {log}\left (\frac {9 x}{9 e^{8}-24 e^{4} x -288 e^{4}+16 x^{2}+384 x +2304}\right )^{2} x +\mathrm {log}\left (\frac {9 x}{9 e^{8}-24 e^{4} x -288 e^{4}+16 x^{2}+384 x +2304}\right ) \mathrm {log}\left (x \right ) x -2 \,\mathrm {log}\left (x \right )}{2 \,\mathrm {log}\left (\frac {9 x}{9 e^{8}-24 e^{4} x -288 e^{4}+16 x^{2}+384 x +2304}\right ) x -4} \] Input:

int(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2 
+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9*x/(9*e 
xp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16*x^3+19 
2*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp 
(4)-16*x^2-192*x),x)
 

Output:

( - 2*log(3*e**4 - 4*x - 48)*log((9*x)/(9*e**8 - 24*e**4*x - 288*e**4 + 16 
*x**2 + 384*x + 2304))*x + 4*log(3*e**4 - 4*x - 48) - log((9*x)/(9*e**8 - 
24*e**4*x - 288*e**4 + 16*x**2 + 384*x + 2304))**2*x + log((9*x)/(9*e**8 - 
 24*e**4*x - 288*e**4 + 16*x**2 + 384*x + 2304))*log(x)*x - 2*log(x))/(2*( 
log((9*x)/(9*e**8 - 24*e**4*x - 288*e**4 + 16*x**2 + 384*x + 2304))*x - 2) 
)