\(\int \frac {390625 \log (x)-x \log ^2(x)+(390625+x \log ^2(x)) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{(-390625 x \log (x)+x^2 \log ^2(x)) \log (x \log (3))+(x+x^2) \log ^2(x) \log ^2(x \log (3))} \, dx\) [1784]

Optimal result

Integrand size = 80, antiderivative size = 20 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (1+x+\frac {x-\frac {390625}{\log (x)}}{\log (x \log (3))}\right ) \] Output:

ln(1+(x-390625/ln(x))/ln(x*ln(3))+x)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx \] Input:

Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3] 
] + x*Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*L 
og[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]
 

Output:

Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3] 
] + x*Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*L 
og[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^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.

\(\displaystyle \int \frac {x \log ^2(x \log (3)) \log ^2(x)-x \log ^2(x)+\left (x \log ^2(x)+390625\right ) \log (x \log (3))+390625 \log (x)}{\left (x^2+x\right ) \log ^2(x) \log ^2(x \log (3))+\left (x^2 \log ^2(x)-390625 x \log (x)\right ) \log (x \log (3))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-x \log ^2(x \log (3)) \log ^2(x)+x \log ^2(x)-\left (x \log ^2(x)+390625\right ) \log (x \log (3))-390625 \log (x)}{x \log (x) \log (x \log (3)) (-x \log (x)-x \log (x \log (3)) \log (x)-\log (x \log (3)) \log (x)+390625)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{x+1}+\frac {390625 x+x \log ^2(x)+390625 x \log (x)+390625}{x (x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625) \log (x)}+\frac {(x+1) \log (x)}{x (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}-\frac {1}{x \log (x \log (3))}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 390625 \int \frac {1}{(x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+390625 \int \frac {1}{x \log (x) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\int \frac {\log (x)}{x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625}dx+\int \frac {\log (x)}{x (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\int \frac {\log (x)}{(x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\log (x+1)-\log (\log (x \log (3)))\)

Input:

Int[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3]] + x* 
Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*Log[3]] 
 + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.62 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95

method	result	size
parallelrisch	\(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \ln \left (3\right )\right )\right )+\ln \left (\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right ) x +\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right )+x \ln \left (x \right )-390625\right )\)	\(39\)
default	\(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (\ln \left (3\right )\right )+\ln \left (x \right )\right )+\ln \left (x \ln \left (x \right ) \ln \left (\ln \left (3\right )\right )+x \ln \left (x \right )^{2}+\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}+x \ln \left (x \right )-390625\right )\)	\(46\)
risch	\(\ln \left (1+x \right )-\ln \left (\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}\right )+\ln \left (\ln \left (x \right )^{2}-\frac {i \left (2 i x \ln \left (\ln \left (3\right )\right )+2 i \ln \left (\ln \left (3\right )\right )+2 i x \right ) \ln \left (x \right )}{2 \left (1+x \right )}-\frac {390625}{1+x}\right )\)	\(61\)

Input:

int((x*ln(x)^2*ln(x*ln(3))^2+(x*ln(x)^2+390625)*ln(x*ln(3))-x*ln(x)^2+3906 
25*ln(x))/((x^2+x)*ln(x)^2*ln(x*ln(3))^2+(x^2*ln(x)^2-390625*x*ln(x))*ln(x 
*ln(3))),x,method=_RETURNVERBOSE)
 

Output:

-ln(ln(x))-ln(ln(x*ln(3)))+ln(ln(x)*ln(x*ln(3))*x+ln(x)*ln(x*ln(3))+x*ln(x 
)-390625)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x + 1\right )} \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* 
log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 
0625*x*log(x))*log(x*log(3))),x, algorithm="fricas")
 

Output:

log(x + 1) + log(((x + 1)*log(x)^2 + (x + 1)*log(x)*log(log(3)) + x*log(x) 
 - 390625)/(x + 1)) - log(log(x) + log(log(3))) - log(log(x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((x*ln(x)**2*ln(x*ln(3))**2+(x*ln(x)**2+390625)*ln(x*ln(3))-x*ln( 
x)**2+390625*ln(x))/((x**2+x)*ln(x)**2*ln(x*ln(3))**2+(x**2*ln(x)**2-39062 
5*x*ln(x))*ln(x*ln(3))),x)
 

Output:

Exception raised: PolynomialError >> 1/(x**6 + 4*x**5 + 6*x**4 + 4*x**3 + 
x**2) contains an element of the set of generators.
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (20) = 40\).

Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.50 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x {\left (\log \left (\log \left (3\right )\right ) + 1\right )} + \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* 
log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 
0625*x*log(x))*log(x*log(3))),x, algorithm="maxima")
 

Output:

log(x + 1) + log(((x + 1)*log(x)^2 + (x*(log(log(3)) + 1) + log(log(3)))*l 
og(x) - 390625)/(x + 1)) - log(log(x) + log(log(3))) - log(log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).

Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x \log \left (x\right )^{2} + x \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) + \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\log \left (3\right )\right ) - 390625\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* 
log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 
0625*x*log(x))*log(x*log(3))),x, algorithm="giac")
 

Output:

log(x*log(x)^2 + x*log(x)*log(log(3)) + x*log(x) + log(x)^2 + log(x)*log(l 
og(3)) - 390625) - log(log(x) + log(log(3))) - log(log(x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625\,\ln \left (x\right )-x\,{\ln \left (x\right )}^2+\ln \left (x\,\ln \left (3\right )\right )\,\left (x\,{\ln \left (x\right )}^2+390625\right )+x\,{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2}{\ln \left (x\,\ln \left (3\right )\right )\,\left (x^2\,{\ln \left (x\right )}^2-390625\,x\,\ln \left (x\right )\right )+{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2\,\left (x^2+x\right )} \,d x \] Input:

int((390625*log(x) - x*log(x)^2 + log(x*log(3))*(x*log(x)^2 + 390625) + x* 
log(x*log(3))^2*log(x)^2)/(log(x*log(3))*(x^2*log(x)^2 - 390625*x*log(x)) 
+ log(x*log(3))^2*log(x)^2*(x + x^2)),x)
 

Output:

int((390625*log(x) - x*log(x)^2 + log(x*log(3))*(x*log(x)^2 + 390625) + x* 
log(x*log(3))^2*log(x)^2)/(log(x*log(3))*(x^2*log(x)^2 - 390625*x*log(x)) 
+ log(x*log(3))^2*log(x)^2*(x + x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right ) x \right )\right )-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (3\right ) x \right ) \mathrm {log}\left (x \right ) x +\mathrm {log}\left (\mathrm {log}\left (3\right ) x \right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x \right ) x -390625\right ) \] Input:

int((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x*log(x) 
^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-390625*x 
*log(x))*log(x*log(3))),x)
 

Output:

 - log(log(log(3)*x)) - log(log(x)) + log(log(log(3)*x)*log(x)*x + log(log 
(3)*x)*log(x) + log(x)*x - 390625)