\(\int \frac {(-15+x^2) \log (x^2)+(180+24 x-12 x^2) \log (\log (x^2))+(-45-3 x^2) \log (x^2) \log ^2(\log (x^2))}{(225-240 x+94 x^2-16 x^3+x^4) \log (x^2)+(1350-540 x-96 x^2+60 x^3-6 x^4) \log (x^2) \log ^2(\log (x^2))+(2025+540 x-234 x^2-36 x^3+9 x^4) \log (x^2) \log ^4(\log (x^2))} \, dx\) [10]

Optimal result

Integrand size = 137, antiderivative size = 26 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{(-5+x) \left (3-x+3 (3+x) \log ^2\left (\log \left (x^2\right )\right )\right )} \] Output:

x/(-5+x)/(3+3*(3+x)*ln(ln(x^2))^2-x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{(-5+x) \left (3-x+3 (3+x) \log ^2\left (\log \left (x^2\right )\right )\right )} \] Input:

Integrate[((-15 + x^2)*Log[x^2] + (180 + 24*x - 12*x^2)*Log[Log[x^2]] + (- 
45 - 3*x^2)*Log[x^2]*Log[Log[x^2]]^2)/((225 - 240*x + 94*x^2 - 16*x^3 + x^ 
4)*Log[x^2] + (1350 - 540*x - 96*x^2 + 60*x^3 - 6*x^4)*Log[x^2]*Log[Log[x^ 
2]]^2 + (2025 + 540*x - 234*x^2 - 36*x^3 + 9*x^4)*Log[x^2]*Log[Log[x^2]]^4 
),x]
 

Output:

x/((-5 + x)*(3 - x + 3*(3 + x)*Log[Log[x^2]]^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.

\(\displaystyle \int \frac {\left (-3 x^2-45\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-12 x^2+24 x+180\right ) \log \left (\log \left (x^2\right )\right )+\left (x^2-15\right ) \log \left (x^2\right )}{\left (9 x^4-36 x^3-234 x^2+540 x+2025\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )+\left (-6 x^4+60 x^3-96 x^2-540 x+1350\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (x^4-16 x^3+94 x^2-240 x+225\right ) \log \left (x^2\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\log \left (x^2\right ) \left (x^2-3 \left (x^2+15\right ) \log ^2\left (\log \left (x^2\right )\right )-15\right )-12 \left (x^2-2 x-15\right ) \log \left (\log \left (x^2\right )\right )}{(5-x)^2 \log \left (x^2\right ) \left (3 (x+3) \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-x^2-15}{(x-5)^2 (x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}-\frac {6 \left (2 x^2 \log \left (\log \left (x^2\right )\right )-x \log \left (x^2\right )+12 x \log \left (\log \left (x^2\right )\right )+18 \log \left (\log \left (x^2\right )\right )\right )}{(x-5) (x+3) \log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {15}{4} \int \frac {1}{(x-5) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx+\frac {9}{4} \int \frac {1}{(x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-12 \int \frac {\log \left (\log \left (x^2\right )\right )}{\log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-96 \int \frac {\log \left (\log \left (x^2\right )\right )}{(x-5) \log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-5 \int \frac {1}{(x-5)^2 \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx-\frac {5}{8} \int \frac {1}{(x-5) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx-\frac {3}{8} \int \frac {1}{(x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx\)

Input:

Int[((-15 + x^2)*Log[x^2] + (180 + 24*x - 12*x^2)*Log[Log[x^2]] + (-45 - 3 
*x^2)*Log[x^2]*Log[Log[x^2]]^2)/((225 - 240*x + 94*x^2 - 16*x^3 + x^4)*Log 
[x^2] + (1350 - 540*x - 96*x^2 + 60*x^3 - 6*x^4)*Log[x^2]*Log[Log[x^2]]^2 
+ (2025 + 540*x - 234*x^2 - 36*x^3 + 9*x^4)*Log[x^2]*Log[Log[x^2]]^4),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(26)=52\).

Time = 5.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.23

method	result	size
parallelrisch	\(-\frac {-45-135 {\ln \left (\ln \left (x^{2}\right )\right )}^{2}-18 {\ln \left (\ln \left (x^{2}\right )\right )}^{2} x +9 {\ln \left (\ln \left (x^{2}\right )\right )}^{2} x^{2}-3 x^{2}}{24 \left (3 {\ln \left (\ln \left (x^{2}\right )\right )}^{2} x^{2}-6 {\ln \left (\ln \left (x^{2}\right )\right )}^{2} x -x^{2}-45 {\ln \left (\ln \left (x^{2}\right )\right )}^{2}+8 x -15\right )}\)	\(84\)

Input:

int(((-3*x^2-45)*ln(x^2)*ln(ln(x^2))^2+(-12*x^2+24*x+180)*ln(ln(x^2))+(x^2 
-15)*ln(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*ln(x^2)*ln(ln(x^2))^4+(-6 
*x^4+60*x^3-96*x^2-540*x+1350)*ln(x^2)*ln(ln(x^2))^2+(x^4-16*x^3+94*x^2-24 
0*x+225)*ln(x^2)),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-45-135*ln(ln(x^2))^2-18*ln(ln(x^2))^2*x+9*ln(ln(x^2))^2*x^2-3*x^2) 
/(3*ln(ln(x^2))^2*x^2-6*ln(ln(x^2))^2*x-x^2-45*ln(ln(x^2))^2+8*x-15)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{3 \, {\left (x^{2} - 2 \, x - 15\right )} \log \left (\log \left (x^{2}\right )\right )^{2} - x^{2} + 8 \, x - 15} \] Input:

integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log 
(x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( 
log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 
4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm="fricas")
 

Output:

x/(3*(x^2 - 2*x - 15)*log(log(x^2))^2 - x^2 + 8*x - 15)
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{- x^{2} + 8 x + \left (3 x^{2} - 6 x - 45\right ) \log {\left (\log {\left (x^{2} \right )} \right )}^{2} - 15} \] Input:

integrate(((-3*x**2-45)*ln(x**2)*ln(ln(x**2))**2+(-12*x**2+24*x+180)*ln(ln 
(x**2))+(x**2-15)*ln(x**2))/((9*x**4-36*x**3-234*x**2+540*x+2025)*ln(x**2) 
*ln(ln(x**2))**4+(-6*x**4+60*x**3-96*x**2-540*x+1350)*ln(x**2)*ln(ln(x**2) 
)**2+(x**4-16*x**3+94*x**2-240*x+225)*ln(x**2)),x)
 

Output:

x/(-x**2 + 8*x + (3*x**2 - 6*x - 45)*log(log(x**2))**2 - 15)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{{\left (3 \, \log \left (2\right )^{2} - 1\right )} x^{2} + 3 \, {\left (x^{2} - 2 \, x - 15\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (3 \, \log \left (2\right )^{2} - 4\right )} x - 45 \, \log \left (2\right )^{2} + 6 \, {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right ) - 15 \, \log \left (2\right )\right )} \log \left (\log \left (x\right )\right ) - 15} \] Input:

integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log 
(x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( 
log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 
4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm="maxima")
 

Output:

x/((3*log(2)^2 - 1)*x^2 + 3*(x^2 - 2*x - 15)*log(log(x))^2 - 2*(3*log(2)^2 
 - 4)*x - 45*log(2)^2 + 6*(x^2*log(2) - 2*x*log(2) - 15*log(2))*log(log(x) 
) - 15)
 

Giac [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{3 \, x^{2} \log \left (\log \left (x^{2}\right )\right )^{2} - 6 \, x \log \left (\log \left (x^{2}\right )\right )^{2} - x^{2} - 45 \, \log \left (\log \left (x^{2}\right )\right )^{2} + 8 \, x - 15} \] Input:

integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log 
(x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( 
log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 
4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm="giac")
 

Output:

x/(3*x^2*log(log(x^2))^2 - 6*x*log(log(x^2))^2 - x^2 - 45*log(log(x^2))^2 
+ 8*x - 15)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\int \frac {-\ln \left (x^2\right )\,\left (3\,x^2+45\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2+\left (-12\,x^2+24\,x+180\right )\,\ln \left (\ln \left (x^2\right )\right )+\ln \left (x^2\right )\,\left (x^2-15\right )}{\ln \left (x^2\right )\,\left (9\,x^4-36\,x^3-234\,x^2+540\,x+2025\right )\,{\ln \left (\ln \left (x^2\right )\right )}^4-\ln \left (x^2\right )\,\left (6\,x^4-60\,x^3+96\,x^2+540\,x-1350\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2+\ln \left (x^2\right )\,\left (x^4-16\,x^3+94\,x^2-240\,x+225\right )} \,d x \] Input:

int((log(log(x^2))*(24*x - 12*x^2 + 180) + log(x^2)*(x^2 - 15) - log(x^2)* 
log(log(x^2))^2*(3*x^2 + 45))/(log(x^2)*(94*x^2 - 240*x - 16*x^3 + x^4 + 2 
25) - log(x^2)*log(log(x^2))^2*(540*x + 96*x^2 - 60*x^3 + 6*x^4 - 1350) + 
log(x^2)*log(log(x^2))^4*(540*x - 234*x^2 - 36*x^3 + 9*x^4 + 2025)),x)
 

Output:

int((log(log(x^2))*(24*x - 12*x^2 + 180) + log(x^2)*(x^2 - 15) - log(x^2)* 
log(log(x^2))^2*(3*x^2 + 45))/(log(x^2)*(94*x^2 - 240*x - 16*x^3 + x^4 + 2 
25) - log(x^2)*log(log(x^2))^2*(540*x + 96*x^2 - 60*x^3 + 6*x^4 - 1350) + 
log(x^2)*log(log(x^2))^4*(540*x - 234*x^2 - 36*x^3 + 9*x^4 + 2025)), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {-3 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2} x^{2}+6 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2} x +45 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2}+x^{2}+15}{24 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2} x^{2}-48 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2} x -360 {\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )}^{2}-8 x^{2}+64 x -120} \] Input:

int(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log(x^2)) 
+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log(log(x^ 
2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^4-16*x 
^3+94*x^2-240*x+225)*log(x^2)),x)
 

Output:

( - 3*log(log(x**2))**2*x**2 + 6*log(log(x**2))**2*x + 45*log(log(x**2))** 
2 + x**2 + 15)/(8*(3*log(log(x**2))**2*x**2 - 6*log(log(x**2))**2*x - 45*l 
og(log(x**2))**2 - x**2 + 8*x - 15))