\(\int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+(-8 x+16 x^2+4 x^6) \log (5)+4 \log ^2(5)} \, dx\) [2689]

Optimal result

Integrand size = 70, antiderivative size = 23 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {3}{-1+2 x+\frac {\frac {x^6}{2}+\log (5)}{x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 x}{-2 x+4 x^2+x^6+\log (25)} \] Input:

Integrate[(-24*x^2 - 30*x^6 + 12*Log[5])/(4*x^2 - 16*x^3 + 16*x^4 - 4*x^7 
+ 8*x^8 + x^12 + (-8*x + 16*x^2 + 4*x^6)*Log[5] + 4*Log[5]^2),x]
 

Output:

(6*x)/(-2*x + 4*x^2 + x^6 + Log[25])
 

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.

\(\displaystyle \int \frac {-30 x^6-24 x^2+12 \log (5)}{x^{12}+8 x^8-4 x^7+16 x^4-16 x^3+4 x^2+\left (4 x^6+16 x^2-8 x\right ) \log (5)+4 \log ^2(5)} \, dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (\frac {12 \left (8 x^2-5 x+6 \log (5)\right )}{\left (x^6+4 x^2-2 x+\log (25)\right )^2}-\frac {30}{x^6+4 x^2-2 x+\log (25)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 72 \log (5) \int \frac {1}{\left (x^6+4 x^2-2 x+\log (25)\right )^2}dx-60 \int \frac {x}{\left (x^6+4 x^2-2 x+\log (25)\right )^2}dx+96 \int \frac {x^2}{\left (x^6+4 x^2-2 x+\log (25)\right )^2}dx-30 \int \frac {1}{x^6+4 x^2-2 x+\log (25)}dx\)

Input:

Int[(-24*x^2 - 30*x^6 + 12*Log[5])/(4*x^2 - 16*x^3 + 16*x^4 - 4*x^7 + 8*x^ 
8 + x^12 + (-8*x + 16*x^2 + 4*x^6)*Log[5] + 4*Log[5]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
gosper	\(\frac {6 x}{x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x}\)	\(22\)
default	\(\frac {3 x}{\frac {x^{6}}{2}+2 x^{2}+\ln \left (5\right )-x}\)	\(22\)
norman	\(\frac {6 x}{x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x}\)	\(22\)
risch	\(\frac {3 x}{\frac {x^{6}}{2}+2 x^{2}+\ln \left (5\right )-x}\)	\(22\)
parallelrisch	\(\frac {6 x}{x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x}\)	\(22\)
orering	\(\frac {\left (x^{6}+4 x^{2}+2 \ln \left (5\right )-2 x \right ) x \left (12 \ln \left (5\right )-30 x^{6}-24 x^{2}\right )}{\left (-5 x^{6}-4 x^{2}+2 \ln \left (5\right )\right ) \left (4 \ln \left (5\right )^{2}+\left (4 x^{6}+16 x^{2}-8 x \right ) \ln \left (5\right )+x^{12}+8 x^{8}-4 x^{7}+16 x^{4}-16 x^{3}+4 x^{2}\right )}\)	\(105\)

Input:

int((12*ln(5)-30*x^6-24*x^2)/(4*ln(5)^2+(4*x^6+16*x^2-8*x)*ln(5)+x^12+8*x^ 
8-4*x^7+16*x^4-16*x^3+4*x^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \] Input:

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+ 
x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 x}{x^{6} + 4 x^{2} - 2 x + 2 \log {\left (5 \right )}} \] Input:

integrate((12*ln(5)-30*x**6-24*x**2)/(4*ln(5)**2+(4*x**6+16*x**2-8*x)*ln(5 
)+x**12+8*x**8-4*x**7+16*x**4-16*x**3+4*x**2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \] Input:

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+ 
x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 \, x}{x^{6} + 4 \, x^{2} - 2 \, x + 2 \, \log \left (5\right )} \] Input:

integrate((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+ 
x^12+8*x^8-4*x^7+16*x^4-16*x^3+4*x^2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 8.82 (sec) , antiderivative size = 1429, normalized size of antiderivative = 62.13 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\text {Too large to display} \] Input:

int(-(24*x^2 - 12*log(5) + 30*x^6)/(log(5)*(16*x^2 - 8*x + 4*x^6) + 4*log( 
5)^2 + 4*x^2 - 16*x^3 + 16*x^4 - 4*x^7 + 8*x^8 + x^12),x)
 

Output:

symsum((log(247669456896) + log(1141803243*log(5) + 761202162*root(7019824 
2049209139200*log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220 
864*log(5)^2 - 178020791885626343424*log(5)^8 + 59817780906615373824*log(5 
)^3 - 328638925913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 44 
926874911493849088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338 
321616896*log(5)^12, z, k)*log(5) - 761202162*root(70198242049209139200*lo 
g(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 1 
78020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 3286389259 
13793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 449268749114938490 
88*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5) 
^12, z, k)*x - 9134425944*x*log(5) - 7806409296*root(70198242049209139200* 
log(5)^9 + 567404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 
 178020791885626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 32863892 
5913793232896*log(5)^4 + 340716998789642059776*log(5)^7 - 4492687491149384 
9088*log(5)^10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log( 
5)^12, z, k)*log(5)^2 + 16542228480*root(70198242049209139200*log(5)^9 + 5 
67404975854624702464*log(5)^5 - 3313942840733220864*log(5)^2 - 17802079188 
5626343424*log(5)^8 + 59817780906615373824*log(5)^3 - 32863892591379323289 
6*log(5)^4 + 340716998789642059776*log(5)^7 - 44926874911493849088*log(5)^ 
10 - 599086626906216333312*log(5)^6 - 4738381338321616896*log(5)^12, z,...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-24 x^2-30 x^6+12 \log (5)}{4 x^2-16 x^3+16 x^4-4 x^7+8 x^8+x^{12}+\left (-8 x+16 x^2+4 x^6\right ) \log (5)+4 \log ^2(5)} \, dx=\frac {6 x}{2 \,\mathrm {log}\left (5\right )+x^{6}+4 x^{2}-2 x} \] Input:

int((12*log(5)-30*x^6-24*x^2)/(4*log(5)^2+(4*x^6+16*x^2-8*x)*log(5)+x^12+8 
*x^8-4*x^7+16*x^4-16*x^3+4*x^2),x)
 

Output:

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