\(\int \frac {30 x^2-10 x^4+(30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)) \log (x)+(18 x^3-12 x^5+2 x^7-120 x^4 \log (3)) \log ^2(x)+(9 x-6 x^3+x^5-60 x^2 \log (3)) \log ^3(x)+((-30-50 x^2+40 x^4) \log (x)+20 x^2 \log ^2(x)) \log (\log (x))}{(9 x^5-6 x^7+x^9) \log (x)+(18 x^3-12 x^5+2 x^7) \log ^2(x)+(9 x-6 x^3+x^5) \log ^3(x)} \, dx\) [1109]

Optimal result

Integrand size = 181, antiderivative size = 28 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=x-\frac {10 \left (-3 \log (3)+\frac {\log (\log (x))}{x^2+\log (x)}\right )}{-3+x^2} \] Output:

x-10/(x^2-3)*(ln(ln(x))/(ln(x)+x^2)-3*ln(3))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=x+\frac {30 \log (3)}{-3+x^2}-\frac {10 \log (\log (x))}{\left (-3+x^2\right ) \left (x^2+\log (x)\right )} \] Input:

Integrate[(30*x^2 - 10*x^4 + (30 - 10*x^2 + 9*x^5 - 6*x^7 + x^9 - 60*x^6*L 
og[3])*Log[x] + (18*x^3 - 12*x^5 + 2*x^7 - 120*x^4*Log[3])*Log[x]^2 + (9*x 
 - 6*x^3 + x^5 - 60*x^2*Log[3])*Log[x]^3 + ((-30 - 50*x^2 + 40*x^4)*Log[x] 
 + 20*x^2*Log[x]^2)*Log[Log[x]])/((9*x^5 - 6*x^7 + x^9)*Log[x] + (18*x^3 - 
 12*x^5 + 2*x^7)*Log[x]^2 + (9*x - 6*x^3 + x^5)*Log[x]^3),x]
 

Output:

x + (30*Log[3])/(-3 + x^2) - (10*Log[Log[x]])/((-3 + x^2)*(x^2 + Log[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.

\(\displaystyle \int \frac {-10 x^4+30 x^2+\left (20 x^2 \log ^2(x)+\left (40 x^4-50 x^2-30\right ) \log (x)\right ) \log (\log (x))+\left (x^5-6 x^3-60 x^2 \log (3)+9 x\right ) \log ^3(x)+\left (2 x^7-12 x^5-120 x^4 \log (3)+18 x^3\right ) \log ^2(x)+\left (x^9-6 x^7-60 x^6 \log (3)+9 x^5-10 x^2+30\right ) \log (x)}{\left (x^5-6 x^3+9 x\right ) \log ^3(x)+\left (x^9-6 x^7+9 x^5\right ) \log (x)+\left (2 x^7-12 x^5+18 x^3\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-10 x^4+30 x^2+\left (20 x^2 \log ^2(x)+\left (40 x^4-50 x^2-30\right ) \log (x)\right ) \log (\log (x))+\left (x^5-6 x^3-60 x^2 \log (3)+9 x\right ) \log ^3(x)+\left (2 x^7-12 x^5-120 x^4 \log (3)+18 x^3\right ) \log ^2(x)+\left (x^9-6 x^7-60 x^6 \log (3)+9 x^5-10 x^2+30\right ) \log (x)}{x \left (3-x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {30 x}{\left (x^2-3\right )^2 \log (x) \left (x^2+\log (x)\right )^2}+\frac {\left (x^4-6 x^2-60 x \log (3)+9\right ) \log ^2(x)}{\left (x^2-3\right )^2 \left (x^2+\log (x)\right )^2}+\frac {2 x^2 \left (x^4-6 x^2-60 x \log (3)+9\right ) \log (x)}{\left (x^2-3\right )^2 \left (x^2+\log (x)\right )^2}+\frac {10 \left (4 x^4-5 x^2+2 x^2 \log (x)-3\right ) \log (\log (x))}{\left (x^2-3\right )^2 x \left (x^2+\log (x)\right )^2}-\frac {10 x^3}{\left (x^2-3\right )^2 \log (x) \left (x^2+\log (x)\right )^2}+\frac {x^9-6 x^7-60 x^6 \log (3)+9 x^5-10 x^2+30}{\left (x^2-3\right )^2 x \left (x^2+\log (x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {30 x}{\left (x^2-3\right )^2 \log (x) \left (x^2+\log (x)\right )^2}+\frac {\left (x^4-6 x^2-60 x \log (3)+9\right ) \log ^2(x)}{\left (x^2-3\right )^2 \left (x^2+\log (x)\right )^2}+\frac {2 x^2 \left (x^4-6 x^2-60 x \log (3)+9\right ) \log (x)}{\left (x^2-3\right )^2 \left (x^2+\log (x)\right )^2}+\frac {10 \left (4 x^4-5 x^2+2 x^2 \log (x)-3\right ) \log (\log (x))}{\left (x^2-3\right )^2 x \left (x^2+\log (x)\right )^2}-\frac {10 x^3}{\left (x^2-3\right )^2 \log (x) \left (x^2+\log (x)\right )^2}+\frac {x^9-6 x^7-60 x^6 \log (3)+9 x^5-10 x^2+30}{\left (x^2-3\right )^2 x \left (x^2+\log (x)\right )^2}\right )dx\)

Input:

Int[(30*x^2 - 10*x^4 + (30 - 10*x^2 + 9*x^5 - 6*x^7 + x^9 - 60*x^6*Log[3]) 
*Log[x] + (18*x^3 - 12*x^5 + 2*x^7 - 120*x^4*Log[3])*Log[x]^2 + (9*x - 6*x 
^3 + x^5 - 60*x^2*Log[3])*Log[x]^3 + ((-30 - 50*x^2 + 40*x^4)*Log[x] + 20* 
x^2*Log[x]^2)*Log[Log[x]])/((9*x^5 - 6*x^7 + x^9)*Log[x] + (18*x^3 - 12*x^ 
5 + 2*x^7)*Log[x]^2 + (9*x - 6*x^3 + x^5)*Log[x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 13.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46

method	result	size
risch	\(-\frac {10 \ln \left (\ln \left (x \right )\right )}{\left (x^{2}-3\right ) \left (\ln \left (x \right )+x^{2}\right )}+\frac {x^{3}+30 \ln \left (3\right )-3 x}{x^{2}-3}\)	\(41\)
parallelrisch	\(\frac {30 x^{2} \ln \left (3\right )-10 \ln \left (\ln \left (x \right )\right )+x^{3} \ln \left (x \right )-3 x \ln \left (x \right )-3 x^{3}+x^{5}+30 \ln \left (x \right ) \ln \left (3\right )}{x^{4}+x^{2} \ln \left (x \right )-3 x^{2}-3 \ln \left (x \right )}\)	\(61\)

Input:

int(((20*x^2*ln(x)^2+(40*x^4-50*x^2-30)*ln(x))*ln(ln(x))+(-60*x^2*ln(3)+x^ 
5-6*x^3+9*x)*ln(x)^3+(-120*x^4*ln(3)+2*x^7-12*x^5+18*x^3)*ln(x)^2+(-60*x^6 
*ln(3)+x^9-6*x^7+9*x^5-10*x^2+30)*ln(x)-10*x^4+30*x^2)/((x^5-6*x^3+9*x)*ln 
(x)^3+(2*x^7-12*x^5+18*x^3)*ln(x)^2+(x^9-6*x^7+9*x^5)*ln(x)),x,method=_RET 
URNVERBOSE)
 

Output:

-10/(x^2-3)/(ln(x)+x^2)*ln(ln(x))+(x^3+30*ln(3)-3*x)/(x^2-3)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=\frac {x^{5} - 3 \, x^{3} + 30 \, x^{2} \log \left (3\right ) + {\left (x^{3} - 3 \, x + 30 \, \log \left (3\right )\right )} \log \left (x\right ) - 10 \, \log \left (\log \left (x\right )\right )}{x^{4} - 3 \, x^{2} + {\left (x^{2} - 3\right )} \log \left (x\right )} \] Input:

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^ 
2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-120*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log 
(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/((x 
^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*lo 
g(x)),x, algorithm="fricas")
 

Output:

(x^5 - 3*x^3 + 30*x^2*log(3) + (x^3 - 3*x + 30*log(3))*log(x) - 10*log(log 
(x)))/(x^4 - 3*x^2 + (x^2 - 3)*log(x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((20*x**2*ln(x)**2+(40*x**4-50*x**2-30)*ln(x))*ln(ln(x))+(-60*x* 
*2*ln(3)+x**5-6*x**3+9*x)*ln(x)**3+(-120*x**4*ln(3)+2*x**7-12*x**5+18*x**3 
)*ln(x)**2+(-60*x**6*ln(3)+x**9-6*x**7+9*x**5-10*x**2+30)*ln(x)-10*x**4+30 
*x**2)/((x**5-6*x**3+9*x)*ln(x)**3+(2*x**7-12*x**5+18*x**3)*ln(x)**2+(x**9 
-6*x**7+9*x**5)*ln(x)),x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=\frac {x^{5} - 3 \, x^{3} + 30 \, x^{2} \log \left (3\right ) + {\left (x^{3} - 3 \, x + 30 \, \log \left (3\right )\right )} \log \left (x\right ) - 10 \, \log \left (\log \left (x\right )\right )}{x^{4} - 3 \, x^{2} + {\left (x^{2} - 3\right )} \log \left (x\right )} \] Input:

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^ 
2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-120*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log 
(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/((x 
^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*lo 
g(x)),x, algorithm="maxima")
 

Output:

(x^5 - 3*x^3 + 30*x^2*log(3) + (x^3 - 3*x + 30*log(3))*log(x) - 10*log(log 
(x)))/(x^4 - 3*x^2 + (x^2 - 3)*log(x))
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=x + \frac {30 \, \log \left (3\right )}{x^{2} - 3} - \frac {10 \, \log \left (\log \left (x\right )\right )}{x^{4} + x^{2} \log \left (x\right ) - 3 \, x^{2} - 3 \, \log \left (x\right )} \] Input:

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^ 
2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-120*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log 
(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/((x 
^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*lo 
g(x)),x, algorithm="giac")
 

Output:

x + 30*log(3)/(x^2 - 3) - 10*log(log(x))/(x^4 + x^2*log(x) - 3*x^2 - 3*log 
(x))
 

Mupad [B] (verification not implemented)

Time = 3.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=x+\frac {30\,x^2\,\ln \left (3\right )-10\,\ln \left (\ln \left (x\right )\right )+30\,\ln \left (3\right )\,\ln \left (x\right )}{\left (\ln \left (x\right )+x^2\right )\,\left (x^2-3\right )} \] Input:

int(-(log(x)*(60*x^6*log(3) + 10*x^2 - 9*x^5 + 6*x^7 - x^9 - 30) + log(x)^ 
2*(120*x^4*log(3) - 18*x^3 + 12*x^5 - 2*x^7) - log(x)^3*(9*x - 60*x^2*log( 
3) - 6*x^3 + x^5) + log(log(x))*(log(x)*(50*x^2 - 40*x^4 + 30) - 20*x^2*lo 
g(x)^2) - 30*x^2 + 10*x^4)/(log(x)^2*(18*x^3 - 12*x^5 + 2*x^7) + log(x)^3* 
(9*x - 6*x^3 + x^5) + log(x)*(9*x^5 - 6*x^7 + x^9)),x)
 

Output:

x + (30*x^2*log(3) - 10*log(log(x)) + 30*log(3)*log(x))/((log(x) + x^2)*(x 
^2 - 3))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx=\frac {-10 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+10 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{2}+\mathrm {log}\left (x \right ) x^{3}-3 \,\mathrm {log}\left (x \right ) x +10 \,\mathrm {log}\left (3\right ) x^{4}+x^{5}-3 x^{3}}{\mathrm {log}\left (x \right ) x^{2}-3 \,\mathrm {log}\left (x \right )+x^{4}-3 x^{2}} \] Input:

int(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^2*log( 
3)+x^5-6*x^3+9*x)*log(x)^3+(-120*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log(x)^2+ 
(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/((x^5-6*x 
^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*log(x)), 
x)
 

Output:

( - 10*log(log(x)) + 10*log(x)*log(3)*x**2 + log(x)*x**3 - 3*log(x)*x + 10 
*log(3)*x**4 + x**5 - 3*x**3)/(log(x)*x**2 - 3*log(x) + x**4 - 3*x**2)