\(\int \frac {64+(80 x^2+40 x^3) \log (\frac {10}{x})+(-40 x^2-40 x^3) \log ^2(\frac {10}{x})}{256+(320 x^2+160 x^3) \log ^2(\frac {10}{x})+(100 x^4+100 x^5+25 x^6) \log ^4(\frac {10}{x})} \, dx\) [2644]

Optimal result

Integrand size = 90, antiderivative size = 24 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {x}{4+\frac {5}{4} x^2 (2+x) \log ^2\left (\frac {10}{x}\right )} \] Output:

x/(4+5/4*x^2*(2+x)*ln(10/x)^2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {8 x}{32+10 x^2 (2+x) \log ^2\left (\frac {10}{x}\right )} \] Input:

Integrate[(64 + (80*x^2 + 40*x^3)*Log[10/x] + (-40*x^2 - 40*x^3)*Log[10/x] 
^2)/(256 + (320*x^2 + 160*x^3)*Log[10/x]^2 + (100*x^4 + 100*x^5 + 25*x^6)* 
Log[10/x]^4),x]
 

Output:

(8*x)/(32 + 10*x^2*(2 + x)*Log[10/x]^2)
 

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.

\(\displaystyle \int \frac {\left (-40 x^3-40 x^2\right ) \log ^2\left (\frac {10}{x}\right )+\left (40 x^3+80 x^2\right ) \log \left (\frac {10}{x}\right )+64}{\left (160 x^3+320 x^2\right ) \log ^2\left (\frac {10}{x}\right )+\left (25 x^6+100 x^5+100 x^4\right ) \log ^4\left (\frac {10}{x}\right )+256} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-40 x^3-40 x^2\right ) \log ^2\left (\frac {10}{x}\right )+\left (40 x^3+80 x^2\right ) \log \left (\frac {10}{x}\right )+64}{\left (5 x^3 \log ^2\left (\frac {10}{x}\right )+10 x^2 \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {8 \left (5 x^4 \log \left (\frac {10}{x}\right )+20 x^3 \log \left (\frac {10}{x}\right )+20 x^2 \log \left (\frac {10}{x}\right )+24 x+32\right )}{(x+2) \left (5 x^3 \log ^2\left (\frac {10}{x}\right )+10 x^2 \log ^2\left (\frac {10}{x}\right )+16\right )^2}-\frac {8 (x+1)}{(x+2) \left (5 x^3 \log ^2\left (\frac {10}{x}\right )+10 x^2 \log ^2\left (\frac {10}{x}\right )+16\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {8 \left (-5 (x+1) x^2 \log ^2\left (\frac {10}{x}\right )+5 (x+2) x^2 \log \left (\frac {10}{x}\right )+8\right )}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {-x-1}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )}+\frac {5 \log \left (\frac {10}{x}\right ) x^4+20 \log \left (\frac {10}{x}\right ) x^3+20 \log \left (\frac {10}{x}\right ) x^2+24 x+32}{(x+2) \left (5 \log ^2\left (\frac {10}{x}\right ) x^3+10 \log ^2\left (\frac {10}{x}\right ) x^2+16\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 8 \int \frac {-5 (x+1) \log ^2\left (\frac {10}{x}\right ) x^2+5 (x+2) \log \left (\frac {10}{x}\right ) x^2+8}{\left (5 x^2 (x+2) \log ^2\left (\frac {10}{x}\right )+16\right )^2}dx\)

Input:

Int[(64 + (80*x^2 + 40*x^3)*Log[10/x] + (-40*x^2 - 40*x^3)*Log[10/x]^2)/(2 
56 + (320*x^2 + 160*x^3)*Log[10/x]^2 + (100*x^4 + 100*x^5 + 25*x^6)*Log[10 
/x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {4 x}{5 \ln \left (\frac {10}{x}\right )^{2} x^{3}+10 x^{2} \ln \left (\frac {10}{x}\right )^{2}+16}\)	\(34\)
parallelrisch	\(\frac {4 x}{5 \ln \left (\frac {10}{x}\right )^{2} x^{3}+10 x^{2} \ln \left (\frac {10}{x}\right )^{2}+16}\)	\(34\)
derivativedivides	\(\frac {500}{x^{2} \left (\frac {2000}{x^{3}}+\frac {1250 \ln \left (\frac {10}{x}\right )^{2}}{x}+625 \ln \left (\frac {10}{x}\right )^{2}\right )}\)	\(37\)
default	\(\frac {500}{x^{2} \left (\frac {2000}{x^{3}}+\frac {1250 \ln \left (\frac {10}{x}\right )^{2}}{x}+625 \ln \left (\frac {10}{x}\right )^{2}\right )}\)	\(37\)

Input:

int(((-40*x^3-40*x^2)*ln(10/x)^2+(40*x^3+80*x^2)*ln(10/x)+64)/((25*x^6+100 
*x^5+100*x^4)*ln(10/x)^4+(160*x^3+320*x^2)*ln(10/x)^2+256),x,method=_RETUR 
NVERBOSE)
 

Output:

4*x/(5*ln(10/x)^2*x^3+10*x^2*ln(10/x)^2+16)
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {4 \, x}{5 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \left (\frac {10}{x}\right )^{2} + 16} \] Input:

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25 
*x^6+100*x^5+100*x^4)*log(10/x)^4+(160*x^3+320*x^2)*log(10/x)^2+256),x, al 
gorithm="fricas")
 

Output:

4*x/(5*(x^3 + 2*x^2)*log(10/x)^2 + 16)
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {4 x}{\left (5 x^{3} + 10 x^{2}\right ) \log {\left (\frac {10}{x} \right )}^{2} + 16} \] Input:

integrate(((-40*x**3-40*x**2)*ln(10/x)**2+(40*x**3+80*x**2)*ln(10/x)+64)/( 
(25*x**6+100*x**5+100*x**4)*ln(10/x)**4+(160*x**3+320*x**2)*ln(10/x)**2+25 
6),x)
 

Output:

4*x/((5*x**3 + 10*x**2)*log(10/x)**2 + 16)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (23) = 46\).

Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {4 \, x}{5 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (2\right ) + \log \left (2\right )^{2}\right )} x^{3} + 10 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (2\right ) + \log \left (2\right )^{2}\right )} x^{2} + 5 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \left (x\right )^{2} - 10 \, {\left (x^{3} {\left (\log \left (5\right ) + \log \left (2\right )\right )} + 2 \, x^{2} {\left (\log \left (5\right ) + \log \left (2\right )\right )}\right )} \log \left (x\right ) + 16} \] Input:

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25 
*x^6+100*x^5+100*x^4)*log(10/x)^4+(160*x^3+320*x^2)*log(10/x)^2+256),x, al 
gorithm="maxima")
 

Output:

4*x/(5*(log(5)^2 + 2*log(5)*log(2) + log(2)^2)*x^3 + 10*(log(5)^2 + 2*log( 
5)*log(2) + log(2)^2)*x^2 + 5*(x^3 + 2*x^2)*log(x)^2 - 10*(x^3*(log(5) + l 
og(2)) + 2*x^2*(log(5) + log(2)))*log(x) + 16)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\frac {4}{{\left (5 \, \log \left (\frac {10}{x}\right )^{2} + \frac {10 \, \log \left (\frac {10}{x}\right )^{2}}{x} + \frac {16}{x^{3}}\right )} x^{2}} \] Input:

integrate(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25 
*x^6+100*x^5+100*x^4)*log(10/x)^4+(160*x^3+320*x^2)*log(10/x)^2+256),x, al 
gorithm="giac")
 

Output:

4/((5*log(10/x)^2 + 10*log(10/x)^2/x + 16/x^3)*x^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=\int \frac {\left (-40\,x^3-40\,x^2\right )\,{\ln \left (\frac {10}{x}\right )}^2+\left (40\,x^3+80\,x^2\right )\,\ln \left (\frac {10}{x}\right )+64}{\left (25\,x^6+100\,x^5+100\,x^4\right )\,{\ln \left (\frac {10}{x}\right )}^4+\left (160\,x^3+320\,x^2\right )\,{\ln \left (\frac {10}{x}\right )}^2+256} \,d x \] Input:

int((log(10/x)*(80*x^2 + 40*x^3) - log(10/x)^2*(40*x^2 + 40*x^3) + 64)/(lo 
g(10/x)^2*(320*x^2 + 160*x^3) + log(10/x)^4*(100*x^4 + 100*x^5 + 25*x^6) + 
 256),x)
 

Output:

int((log(10/x)*(80*x^2 + 40*x^3) - log(10/x)^2*(40*x^2 + 40*x^3) + 64)/(lo 
g(10/x)^2*(320*x^2 + 160*x^3) + log(10/x)^4*(100*x^4 + 100*x^5 + 25*x^6) + 
 256), x)
 

Reduce [F]

\[ \int \frac {64+\left (80 x^2+40 x^3\right ) \log \left (\frac {10}{x}\right )+\left (-40 x^2-40 x^3\right ) \log ^2\left (\frac {10}{x}\right )}{256+\left (320 x^2+160 x^3\right ) \log ^2\left (\frac {10}{x}\right )+\left (100 x^4+100 x^5+25 x^6\right ) \log ^4\left (\frac {10}{x}\right )} \, dx=-40 \left (\int \frac {\mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}}{25 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{6}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{5}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{4}+160 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}+320 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}+256}d x \right )-40 \left (\int \frac {\mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}}{25 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{6}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{5}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{4}+160 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}+320 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}+256}d x \right )+40 \left (\int \frac {\mathrm {log}\left (\frac {10}{x}\right ) x^{3}}{25 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{6}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{5}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{4}+160 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}+320 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}+256}d x \right )+80 \left (\int \frac {\mathrm {log}\left (\frac {10}{x}\right ) x^{2}}{25 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{6}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{5}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{4}+160 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}+320 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}+256}d x \right )+64 \left (\int \frac {1}{25 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{6}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{5}+100 \mathrm {log}\left (\frac {10}{x}\right )^{4} x^{4}+160 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{3}+320 \mathrm {log}\left (\frac {10}{x}\right )^{2} x^{2}+256}d x \right ) \] Input:

int(((-40*x^3-40*x^2)*log(10/x)^2+(40*x^3+80*x^2)*log(10/x)+64)/((25*x^6+1 
00*x^5+100*x^4)*log(10/x)^4+(160*x^3+320*x^2)*log(10/x)^2+256),x)
 

Output:

8*( - 5*int((log(10/x)**2*x**3)/(25*log(10/x)**4*x**6 + 100*log(10/x)**4*x 
**5 + 100*log(10/x)**4*x**4 + 160*log(10/x)**2*x**3 + 320*log(10/x)**2*x** 
2 + 256),x) - 5*int((log(10/x)**2*x**2)/(25*log(10/x)**4*x**6 + 100*log(10 
/x)**4*x**5 + 100*log(10/x)**4*x**4 + 160*log(10/x)**2*x**3 + 320*log(10/x 
)**2*x**2 + 256),x) + 5*int((log(10/x)*x**3)/(25*log(10/x)**4*x**6 + 100*l 
og(10/x)**4*x**5 + 100*log(10/x)**4*x**4 + 160*log(10/x)**2*x**3 + 320*log 
(10/x)**2*x**2 + 256),x) + 10*int((log(10/x)*x**2)/(25*log(10/x)**4*x**6 + 
 100*log(10/x)**4*x**5 + 100*log(10/x)**4*x**4 + 160*log(10/x)**2*x**3 + 3 
20*log(10/x)**2*x**2 + 256),x) + 8*int(1/(25*log(10/x)**4*x**6 + 100*log(1 
0/x)**4*x**5 + 100*log(10/x)**4*x**4 + 160*log(10/x)**2*x**3 + 320*log(10/ 
x)**2*x**2 + 256),x))