\(\int \frac {-41 x+x^2-120 x^3-72 x^5+(-19 x^2-24 x^4) \log (\frac {36}{x})-2 x^3 \log ^2(\frac {36}{x})+(-18 x-24 x^3-4 x^2 \log (\frac {36}{x})) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+(10 x+12 x^3) \log (\frac {36}{x})+x^2 \log ^2(\frac {36}{x})+(10+12 x^2+2 x \log (\frac {36}{x})) \log (x)+\log ^2(x)} \, dx\) [842]

Optimal result

Integrand size = 144, antiderivative size = 26 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=x \left (-x+\frac {x}{5+x \left (6 x+\log \left (\frac {36}{x}\right )\right )+\log (x)}\right ) \] Output:

x*(x/(ln(x)+5+x*(6*x+ln(36/x)))-x)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-x^2+\frac {x^2}{5+6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)} \] Input:

Integrate[(-41*x + x^2 - 120*x^3 - 72*x^5 + (-19*x^2 - 24*x^4)*Log[36/x] - 
 2*x^3*Log[36/x]^2 + (-18*x - 24*x^3 - 4*x^2*Log[36/x])*Log[x] - 2*x*Log[x 
]^2)/(25 + 60*x^2 + 36*x^4 + (10*x + 12*x^3)*Log[36/x] + x^2*Log[36/x]^2 + 
 (10 + 12*x^2 + 2*x*Log[36/x])*Log[x] + Log[x]^2),x]
 

Output:

-x^2 + x^2/(5 + 6*x^2 + x*Log[36/x] + 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.

\(\displaystyle \int \frac {-72 x^5-120 x^3-2 x^3 \log ^2\left (\frac {36}{x}\right )+x^2+\left (-24 x^4-19 x^2\right ) \log \left (\frac {36}{x}\right )+\left (-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )-18 x\right ) \log (x)-41 x-2 x \log ^2(x)}{36 x^4+\left (12 x^3+10 x\right ) \log \left (\frac {36}{x}\right )+60 x^2+x^2 \log ^2\left (\frac {36}{x}\right )+\left (12 x^2+2 x \log \left (\frac {36}{x}\right )+10\right ) \log (x)+\log ^2(x)+25} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-72 x^4-120 x^2-2 x^2 \log ^2\left (\frac {36}{x}\right )-x \log \left (\frac {36}{x}\right ) \left (24 x^2+4 \log (x)+19\right )-6 \left (4 x^2+3\right ) \log (x)+x-2 \log ^2(x)-41\right )}{\left (6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x}{6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)+5}-\frac {x \left (12 x^2-x+x \log \left (\frac {36}{x}\right )+1\right )}{\left (6 x^2+x \log \left (\frac {36}{x}\right )+\log (x)+5\right )^2}-2 x\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {x}{\left (6 x^2+\log \left (\frac {36}{x}\right ) x+\log (x)+5\right )^2}dx+\int \frac {x^2}{\left (6 x^2+\log \left (\frac {36}{x}\right ) x+\log (x)+5\right )^2}dx-\int \frac {x^2 \log \left (\frac {36}{x}\right )}{\left (6 x^2+\log \left (\frac {36}{x}\right ) x+\log (x)+5\right )^2}dx+2 \int \frac {x}{6 x^2+\log \left (\frac {36}{x}\right ) x+\log (x)+5}dx-12 \int \frac {x^3}{\left (6 x^2+\log \left (\frac {36}{x}\right ) x+\log (x)+5\right )^2}dx-x^2\)

Input:

Int[(-41*x + x^2 - 120*x^3 - 72*x^5 + (-19*x^2 - 24*x^4)*Log[36/x] - 2*x^3 
*Log[36/x]^2 + (-18*x - 24*x^3 - 4*x^2*Log[36/x])*Log[x] - 2*x*Log[x]^2)/( 
25 + 60*x^2 + 36*x^4 + (10*x + 12*x^3)*Log[36/x] + x^2*Log[36/x]^2 + (10 + 
 12*x^2 + 2*x*Log[36/x])*Log[x] + Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.40 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54

method	result	size
risch	\(-x^{2}+\frac {2 x^{2}}{10+4 x \ln \left (3\right )+4 x \ln \left (2\right )+12 x^{2}-2 x \ln \left (x \right )+2 \ln \left (x \right )}\)	\(40\)
parallelrisch	\(\frac {-x^{2} \ln \left (x \right )-4 x^{2}-6 x^{4}-x^{3} \ln \left (\frac {36}{x}\right )}{x \ln \left (\frac {36}{x}\right )+6 x^{2}+\ln \left (x \right )+5}\)	\(50\)
default	\(\frac {\left (-6+\frac {\ln \left (\frac {1}{x}\right )}{x^{2}}+\frac {-4-\ln \left (\frac {1}{x}\right )-\ln \left (x \right )}{x^{2}}-\frac {\ln \left (\frac {1}{x}\right )}{x}-\frac {2 \ln \left (6\right )}{x}\right ) x^{2}}{-\frac {\ln \left (\frac {1}{x}\right )}{x^{2}}+6+\frac {\ln \left (x \right )+\ln \left (\frac {1}{x}\right )}{x^{2}}+\frac {2 \ln \left (6\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+\frac {5}{x^{2}}}\)	\(91\)

Input:

int((-2*x*ln(x)^2+(-4*x^2*ln(36/x)-24*x^3-18*x)*ln(x)-2*x^3*ln(36/x)^2+(-2 
4*x^4-19*x^2)*ln(36/x)-72*x^5-120*x^3+x^2-41*x)/(ln(x)^2+(2*x*ln(36/x)+12* 
x^2+10)*ln(x)+x^2*ln(36/x)^2+(12*x^3+10*x)*ln(36/x)+36*x^4+60*x^2+25),x,me 
thod=_RETURNVERBOSE)
 

Output:

-x^2+2*x^2/(10+4*x*ln(3)+4*x*ln(2)+12*x^2-2*x*ln(x)+2*ln(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {6 \, x^{4} + 2 \, x^{2} \log \left (6\right ) + 4 \, x^{2} + {\left (x^{3} - x^{2}\right )} \log \left (\frac {36}{x}\right )}{6 \, x^{2} + {\left (x - 1\right )} \log \left (\frac {36}{x}\right ) + 2 \, \log \left (6\right ) + 5} \] Input:

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(3 
6/x)^2+(-24*x^4-19*x^2)*log(36/x)-72*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x* 
log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x^4 
+60*x^2+25),x, algorithm="fricas")
 

Output:

-(6*x^4 + 2*x^2*log(6) + 4*x^2 + (x^3 - x^2)*log(36/x))/(6*x^2 + (x - 1)*l 
og(36/x) + 2*log(6) + 5)
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=- x^{2} - \frac {x^{2}}{- 6 x^{2} - 2 x \log {\left (6 \right )} + \left (x - 1\right ) \log {\left (x \right )} - 5} \] Input:

integrate((-2*x*ln(x)**2+(-4*x**2*ln(36/x)-24*x**3-18*x)*ln(x)-2*x**3*ln(3 
6/x)**2+(-24*x**4-19*x**2)*ln(36/x)-72*x**5-120*x**3+x**2-41*x)/(ln(x)**2+ 
(2*x*ln(36/x)+12*x**2+10)*ln(x)+x**2*ln(36/x)**2+(12*x**3+10*x)*ln(36/x)+3 
6*x**4+60*x**2+25),x)
 

Output:

-x**2 - x**2/(-6*x**2 - 2*x*log(6) + (x - 1)*log(x) - 5)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {6 \, x^{4} + 2 \, x^{3} {\left (\log \left (3\right ) + \log \left (2\right )\right )} + 4 \, x^{2} - {\left (x^{3} - x^{2}\right )} \log \left (x\right )}{6 \, x^{2} + 2 \, x {\left (\log \left (3\right ) + \log \left (2\right )\right )} - {\left (x - 1\right )} \log \left (x\right ) + 5} \] Input:

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(3 
6/x)^2+(-24*x^4-19*x^2)*log(36/x)-72*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x* 
log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x^4 
+60*x^2+25),x, algorithm="maxima")
 

Output:

-(6*x^4 + 2*x^3*(log(3) + log(2)) + 4*x^2 - (x^3 - x^2)*log(x))/(6*x^2 + 2 
*x*(log(3) + log(2)) - (x - 1)*log(x) + 5)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-x^{2} + \frac {x^{2}}{6 \, x^{2} + 2 \, x \log \left (6\right ) - x \log \left (x\right ) + \log \left (x\right ) + 5} \] Input:

integrate((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(3 
6/x)^2+(-24*x^4-19*x^2)*log(36/x)-72*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x* 
log(36/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x^4 
+60*x^2+25),x, algorithm="giac")
 

Output:

-x^2 + x^2/(6*x^2 + 2*x*log(6) - x*log(x) + log(x) + 5)
 

Mupad [B] (verification not implemented)

Time = 3.63 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=-\frac {x^2\,\left (\ln \left (x\right )+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+4\right )}{\ln \left (x\right )+x\,\ln \left (\frac {36}{x}\right )+6\,x^2+5} \] Input:

int(-(41*x + 2*x*log(x)^2 + 2*x^3*log(36/x)^2 + log(x)*(18*x + 24*x^3 + 4* 
x^2*log(36/x)) + log(36/x)*(19*x^2 + 24*x^4) - x^2 + 120*x^3 + 72*x^5)/(x^ 
2*log(36/x)^2 + log(x)^2 + log(x)*(2*x*log(36/x) + 12*x^2 + 10) + log(36/x 
)*(10*x + 12*x^3) + 60*x^2 + 36*x^4 + 25),x)
 

Output:

-(x^2*(log(x) + x*log(36/x) + 6*x^2 + 4))/(log(x) + x*log(36/x) + 6*x^2 + 
5)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-41 x+x^2-120 x^3-72 x^5+\left (-19 x^2-24 x^4\right ) \log \left (\frac {36}{x}\right )-2 x^3 \log ^2\left (\frac {36}{x}\right )+\left (-18 x-24 x^3-4 x^2 \log \left (\frac {36}{x}\right )\right ) \log (x)-2 x \log ^2(x)}{25+60 x^2+36 x^4+\left (10 x+12 x^3\right ) \log \left (\frac {36}{x}\right )+x^2 \log ^2\left (\frac {36}{x}\right )+\left (10+12 x^2+2 x \log \left (\frac {36}{x}\right )\right ) \log (x)+\log ^2(x)} \, dx=\frac {x^{2} \left (-\mathrm {log}\left (\frac {36}{x}\right ) x -\mathrm {log}\left (x \right )-6 x^{2}-4\right )}{\mathrm {log}\left (\frac {36}{x}\right ) x +\mathrm {log}\left (x \right )+6 x^{2}+5} \] Input:

int((-2*x*log(x)^2+(-4*x^2*log(36/x)-24*x^3-18*x)*log(x)-2*x^3*log(36/x)^2 
+(-24*x^4-19*x^2)*log(36/x)-72*x^5-120*x^3+x^2-41*x)/(log(x)^2+(2*x*log(36 
/x)+12*x^2+10)*log(x)+x^2*log(36/x)^2+(12*x^3+10*x)*log(36/x)+36*x^4+60*x^ 
2+25),x)
 

Output:

(x**2*( - log(36/x)*x - log(x) - 6*x**2 - 4))/(log(36/x)*x + log(x) + 6*x* 
*2 + 5)