\(\int \frac {(-30-12 x^2-6 x^3+6 x^4) \log ^2(\frac {5+2 x^2+x^3-x^4}{x^2})+\log (\frac {2}{x}) ((60-6 x^3+12 x^4) \log (\frac {5+2 x^2+x^3-x^4}{x^2})+(30+12 x^2+6 x^3-6 x^4) \log ^2(\frac {5+2 x^2+x^3-x^4}{x^2}))}{(-5 x^3-2 x^5-x^6+x^7) \log ^3(\frac {2}{x})} \, dx\) [2647]

Optimal result

Integrand size = 152, antiderivative size = 29 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3 \log ^2\left (2+\frac {5}{x^2}+x-x^2\right )}{x^2 \log ^2\left (\frac {2}{x}\right )} \] Output:

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

Mathematica [F]

\[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx \] Input:

Integrate[((-30 - 12*x^2 - 6*x^3 + 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2] 
^2 + Log[2/x]*((60 - 6*x^3 + 12*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2] + (3 
0 + 12*x^2 + 6*x^3 - 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2]^2))/((-5*x^3 
- 2*x^5 - x^6 + x^7)*Log[2/x]^3),x]
 

Output:

Integrate[((-30 - 12*x^2 - 6*x^3 + 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2] 
^2 + Log[2/x]*((60 - 6*x^3 + 12*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2] + (3 
0 + 12*x^2 + 6*x^3 - 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2]^2))/((-5*x^3 
- 2*x^5 - x^6 + x^7)*Log[2/x]^3), 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.

\(\displaystyle \int \frac {\left (6 x^4-6 x^3-12 x^2-30\right ) \log ^2\left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (-6 x^4+6 x^3+12 x^2+30\right ) \log ^2\left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )+\left (12 x^4-6 x^3+60\right ) \log \left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )\right )}{\left (x^7-x^6-2 x^5-5 x^3\right ) \log ^3\left (\frac {2}{x}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (6 x^4-6 x^3-12 x^2-30\right ) \log ^2\left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (-6 x^4+6 x^3+12 x^2+30\right ) \log ^2\left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )+\left (12 x^4-6 x^3+60\right ) \log \left (\frac {-x^4+x^3+2 x^2+5}{x^2}\right )\right )}{x^3 \left (x^4-x^3-2 x^2-5\right ) \log ^3\left (\frac {2}{x}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {6 \left (2 x^4-x^3+10\right ) \log \left (-x^2+\frac {5}{x^2}+x+2\right )}{x^3 \left (x^4-x^3-2 x^2-5\right ) \log ^2\left (\frac {2}{x}\right )}-\frac {6 \left (\log \left (\frac {2}{x}\right )-1\right ) \log ^2\left (-x^2+\frac {5}{x^2}+x+2\right )}{x^3 \log ^3\left (\frac {2}{x}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {24}{5} \int \frac {\log \left (-x^2+x+2+\frac {5}{x^2}\right )}{x \log ^2\left (\frac {2}{x}\right )}dx-12 \int \frac {\log \left (-x^2+x+2+\frac {5}{x^2}\right )}{x^3 \log ^2\left (\frac {2}{x}\right )}dx-6 \int \frac {\log ^2\left (-x^2+x+2+\frac {5}{x^2}\right )}{x^3 \log ^2\left (\frac {2}{x}\right )}dx+6 \int \frac {\log ^2\left (-x^2+x+2+\frac {5}{x^2}\right )}{x^3 \log ^3\left (\frac {2}{x}\right )}dx-18 \int \frac {\log \left (-x^2+x+2+\frac {5}{x^2}\right )}{\left (x^4-x^3-2 x^2-5\right ) \log ^2\left (\frac {2}{x}\right )}dx+\frac {168}{5} \int \frac {x \log \left (-x^2+x+2+\frac {5}{x^2}\right )}{\left (x^4-x^3-2 x^2-5\right ) \log ^2\left (\frac {2}{x}\right )}dx+\frac {24}{5} \int \frac {x^2 \log \left (-x^2+x+2+\frac {5}{x^2}\right )}{\left (x^4-x^3-2 x^2-5\right ) \log ^2\left (\frac {2}{x}\right )}dx-\frac {24}{5} \int \frac {x^3 \log \left (-x^2+x+2+\frac {5}{x^2}\right )}{\left (x^4-x^3-2 x^2-5\right ) \log ^2\left (\frac {2}{x}\right )}dx\)

Input:

Int[((-30 - 12*x^2 - 6*x^3 + 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2]^2 + L 
og[2/x]*((60 - 6*x^3 + 12*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2] + (30 + 12 
*x^2 + 6*x^3 - 6*x^4)*Log[(5 + 2*x^2 + x^3 - x^4)/x^2]^2))/((-5*x^3 - 2*x^ 
5 - x^6 + x^7)*Log[2/x]^3),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.50 (sec) , antiderivative size = 2291, normalized size of antiderivative = 79.00

\[\text {Expression too large to display}\]

Input:

int((((-6*x^4+6*x^3+12*x^2+30)*ln((-x^4+x^3+2*x^2+5)/x^2)^2+(12*x^4-6*x^3+ 
60)*ln((-x^4+x^3+2*x^2+5)/x^2))*ln(2/x)+(6*x^4-6*x^3-12*x^2-30)*ln((-x^4+x 
^3+2*x^2+5)/x^2)^2)/(x^7-x^6-2*x^5-5*x^3)/ln(2/x)^3,x)
 

Output:

-12/x^2/(2*I*ln(2)-2*I*ln(x))^2*ln(x^4-x^3-2*x^2-5)^2-12*(-2*I*Pi+4*ln(x)- 
I*Pi*csgn(I*x^2)^3+I*Pi*csgn(I/x^2)*csgn(I*(x^4-x^3-2*x^2-5))*csgn(I/x^2*( 
x^4-x^3-2*x^2-5))+2*I*Pi*csgn(I/x^2*(x^4-x^3-2*x^2-5))^2-I*Pi*csgn(I*x)^2* 
csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I/x^2)*csgn(I/x^2*(x^ 
4-x^3-2*x^2-5))^2-I*Pi*csgn(I*(x^4-x^3-2*x^2-5))*csgn(I/x^2*(x^4-x^3-2*x^2 
-5))^2-I*Pi*csgn(I/x^2*(x^4-x^3-2*x^2-5))^3)/x^2/(-2*ln(2)+2*ln(x))^2*ln(x 
^4-x^3-2*x^2-5)+3*(8*Pi^2*csgn(I/x^2*(x^4-x^3-2*x^2-5))^2-Pi^2*csgn(I/x^2* 
(x^4-x^3-2*x^2-5))^6-4*Pi^2*csgn(I*x^2)^3-4*Pi^2+16*ln(x)^2-Pi^2*csgn(I*x) 
^4*csgn(I*x^2)^2+4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-6*Pi^2*csgn(I*x)^2*csgn( 
I*x^2)^4+4*Pi^2*csgn(I*x)*csgn(I*x^2)^5-Pi^2*csgn(I*x^2)^6+4*Pi^2*csgn(I/x 
^2)*csgn(I/x^2*(x^4-x^3-2*x^2-5))^4+4*Pi^2*csgn(I*(x^4-x^3-2*x^2-5))*csgn( 
I/x^2*(x^4-x^3-2*x^2-5))^4-4*Pi^2*csgn(I*x)^2*csgn(I*x^2)+8*Pi^2*csgn(I*x) 
*csgn(I*x^2)^2-4*Pi^2*csgn(I/x^2)*csgn(I/x^2*(x^4-x^3-2*x^2-5))^2-4*Pi^2*c 
sgn(I*(x^4-x^3-2*x^2-5))*csgn(I/x^2*(x^4-x^3-2*x^2-5))^2-2*Pi^2*csgn(I*x^2 
)^3*csgn(I/x^2*(x^4-x^3-2*x^2-5))^3-Pi^2*csgn(I/x^2)^2*csgn(I/x^2*(x^4-x^3 
-2*x^2-5))^4-2*Pi^2*csgn(I/x^2)*csgn(I/x^2*(x^4-x^3-2*x^2-5))^5-Pi^2*csgn( 
I*(x^4-x^3-2*x^2-5))^2*csgn(I/x^2*(x^4-x^3-2*x^2-5))^4-2*Pi^2*csgn(I*(x^4- 
x^3-2*x^2-5))*csgn(I/x^2*(x^4-x^3-2*x^2-5))^5-16*I*Pi*ln(x)-4*Pi^2*csgn(I/ 
x^2*(x^4-x^3-2*x^2-5))^3-4*Pi^2*csgn(I/x^2*(x^4-x^3-2*x^2-5))^4+4*Pi^2*csg 
n(I/x^2*(x^4-x^3-2*x^2-5))^5-8*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I/x^2*...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3 \, \log \left (-\frac {x^{4} - x^{3} - 2 \, x^{2} - 5}{x^{2}}\right )^{2}}{x^{2} \log \left (\frac {2}{x}\right )^{2}} \] Input:

integrate((((-6*x^4+6*x^3+12*x^2+30)*log((-x^4+x^3+2*x^2+5)/x^2)^2+(12*x^4 
-6*x^3+60)*log((-x^4+x^3+2*x^2+5)/x^2))*log(2/x)+(6*x^4-6*x^3-12*x^2-30)*l 
og((-x^4+x^3+2*x^2+5)/x^2)^2)/(x^7-x^6-2*x^5-5*x^3)/log(2/x)^3,x, algorith 
m="fricas")
 

Output:

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

Sympy [F(-2)]

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

integrate((((-6*x**4+6*x**3+12*x**2+30)*ln((-x**4+x**3+2*x**2+5)/x**2)**2+ 
(12*x**4-6*x**3+60)*ln((-x**4+x**3+2*x**2+5)/x**2))*ln(2/x)+(6*x**4-6*x**3 
-12*x**2-30)*ln((-x**4+x**3+2*x**2+5)/x**2)**2)/(x**7-x**6-2*x**5-5*x**3)/ 
ln(2/x)**3,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3 \, {\left (\log \left (-x^{4} + x^{3} + 2 \, x^{2} + 5\right )^{2} - 4 \, \log \left (-x^{4} + x^{3} + 2 \, x^{2} + 5\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}\right )}}{x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} \] Input:

integrate((((-6*x^4+6*x^3+12*x^2+30)*log((-x^4+x^3+2*x^2+5)/x^2)^2+(12*x^4 
-6*x^3+60)*log((-x^4+x^3+2*x^2+5)/x^2))*log(2/x)+(6*x^4-6*x^3-12*x^2-30)*l 
og((-x^4+x^3+2*x^2+5)/x^2)^2)/(x^7-x^6-2*x^5-5*x^3)/log(2/x)^3,x, algorith 
m="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (29) = 58\).

Time = 0.79 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.93 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3 \, \log \left (-x^{4} + x^{3} + 2 \, x^{2} + 5\right )^{2}}{x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} - \frac {12 \, \log \left (-x^{4} + x^{3} + 2 \, x^{2} + 5\right ) \log \left (x\right )}{x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} - \frac {12 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) \log \left (x\right )\right )}}{x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2}} + \frac {12}{x^{2}} \] Input:

integrate((((-6*x^4+6*x^3+12*x^2+30)*log((-x^4+x^3+2*x^2+5)/x^2)^2+(12*x^4 
-6*x^3+60)*log((-x^4+x^3+2*x^2+5)/x^2))*log(2/x)+(6*x^4-6*x^3-12*x^2-30)*l 
og((-x^4+x^3+2*x^2+5)/x^2)^2)/(x^7-x^6-2*x^5-5*x^3)/log(2/x)^3,x, algorith 
m="giac")
 

Output:

3*log(-x^4 + x^3 + 2*x^2 + 5)^2/(x^2*log(2)^2 - 2*x^2*log(2)*log(x) + x^2* 
log(x)^2) - 12*log(-x^4 + x^3 + 2*x^2 + 5)*log(x)/(x^2*log(2)^2 - 2*x^2*lo 
g(2)*log(x) + x^2*log(x)^2) - 12*(log(2)^2 - 2*log(2)*log(x))/(x^2*log(2)^ 
2 - 2*x^2*log(2)*log(x) + x^2*log(x)^2) + 12/x^2
 

Mupad [B] (verification not implemented)

Time = 3.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3\,{\ln \left (\frac {-x^4+x^3+2\,x^2+5}{x^2}\right )}^2}{x^2\,{\ln \left (\frac {2}{x}\right )}^2} \] Input:

int((log((2*x^2 + x^3 - x^4 + 5)/x^2)^2*(12*x^2 + 6*x^3 - 6*x^4 + 30) - lo 
g(2/x)*(log((2*x^2 + x^3 - x^4 + 5)/x^2)^2*(12*x^2 + 6*x^3 - 6*x^4 + 30) + 
 log((2*x^2 + x^3 - x^4 + 5)/x^2)*(12*x^4 - 6*x^3 + 60)))/(log(2/x)^3*(5*x 
^3 + 2*x^5 + x^6 - x^7)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-30-12 x^2-6 x^3+6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\log \left (\frac {2}{x}\right ) \left (\left (60-6 x^3+12 x^4\right ) \log \left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )+\left (30+12 x^2+6 x^3-6 x^4\right ) \log ^2\left (\frac {5+2 x^2+x^3-x^4}{x^2}\right )\right )}{\left (-5 x^3-2 x^5-x^6+x^7\right ) \log ^3\left (\frac {2}{x}\right )} \, dx=\frac {3 \mathrm {log}\left (\frac {-x^{4}+x^{3}+2 x^{2}+5}{x^{2}}\right )^{2}}{\mathrm {log}\left (\frac {2}{x}\right )^{2} x^{2}} \] Input:

int((((-6*x^4+6*x^3+12*x^2+30)*log((-x^4+x^3+2*x^2+5)/x^2)^2+(12*x^4-6*x^3 
+60)*log((-x^4+x^3+2*x^2+5)/x^2))*log(2/x)+(6*x^4-6*x^3-12*x^2-30)*log((-x 
^4+x^3+2*x^2+5)/x^2)^2)/(x^7-x^6-2*x^5-5*x^3)/log(2/x)^3,x)
 

Output:

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