\(\int \frac {-75-75 x-x^2+x^3+2 x^4+((75-x^2+x^3) \log (2)+(-75+x^2-x^3) \log (\frac {75-x^2+x^3}{x})) \log (\frac {-\log (2)+\log (\frac {75-x^2+x^3}{x})}{\log (2)})}{(-75 x^2+x^4-x^5) \log (2)+(75 x^2-x^4+x^5) \log (\frac {75-x^2+x^3}{x})} \, dx\) [3]

Optimal result

Integrand size = 136, antiderivative size = 31 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=5+\frac {(1+x) \log \left (-1+\frac {\log \left (-x+x \left (\frac {75}{x^2}+x\right )\right )}{\log (2)}\right )}{x} \] Output:

ln(ln(x*(75/x^2+x)-x)/ln(2)-1)/x*(1+x)+5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\log \left (\log (2)-\log \left (\frac {75}{x}-x+x^2\right )\right )+\frac {\log \left (-1+\frac {\log \left (\frac {75}{x}-x+x^2\right )}{\log (2)}\right )}{x} \] Input:

Integrate[(-75 - 75*x - x^2 + x^3 + 2*x^4 + ((75 - x^2 + x^3)*Log[2] + (-7 
5 + x^2 - x^3)*Log[(75 - x^2 + x^3)/x])*Log[(-Log[2] + Log[(75 - x^2 + x^3 
)/x])/Log[2]])/((-75*x^2 + x^4 - x^5)*Log[2] + (75*x^2 - x^4 + x^5)*Log[(7 
5 - x^2 + x^3)/x]),x]
 

Output:

Log[Log[2] - Log[75/x - x + x^2]] + Log[-1 + Log[75/x - x + x^2]/Log[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.

\(\displaystyle \int \frac {2 x^4+x^3-x^2+\left (\left (x^3-x^2+75\right ) \log (2)+\left (-x^3+x^2-75\right ) \log \left (\frac {x^3-x^2+75}{x}\right )\right ) \log \left (\frac {\log \left (\frac {x^3-x^2+75}{x}\right )-\log (2)}{\log (2)}\right )-75 x-75}{\left (-x^5+x^4-75 x^2\right ) \log (2)+\left (x^5-x^4+75 x^2\right ) \log \left (\frac {x^3-x^2+75}{x}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-2 x^4-x^3+x^2-\left (\left (x^3-x^2+75\right ) \log (2)+\left (-x^3+x^2-75\right ) \log \left (\frac {x^3-x^2+75}{x}\right )\right ) \log \left (\frac {\log \left (\frac {x^3-x^2+75}{x}\right )-\log (2)}{\log (2)}\right )+75 x+75}{x^2 \left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {\log \left (\frac {\log \left (x^2-x+\frac {75}{x}\right )}{\log (2)}-1\right )}{x^2}-\frac {2 x^2}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}-\frac {x}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}+\frac {1}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}+\frac {75}{\left (x^3-x^2+75\right ) x \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}+\frac {75}{\left (x^3-x^2+75\right ) x^2 \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{x^2 \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx+\int \frac {1}{x \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx-\int \frac {\log \left (\frac {\log \left (x^2-x+\frac {75}{x}\right )}{\log (2)}-1\right )}{x^2}dx+2 \int \frac {1}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx-\int \frac {x}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx-3 \int \frac {x^2}{\left (x^3-x^2+75\right ) \left (\log (2)-\log \left (x^2-x+\frac {75}{x}\right )\right )}dx\)

Input:

Int[(-75 - 75*x - x^2 + x^3 + 2*x^4 + ((75 - x^2 + x^3)*Log[2] + (-75 + x^ 
2 - x^3)*Log[(75 - x^2 + x^3)/x])*Log[(-Log[2] + Log[(75 - x^2 + x^3)/x])/ 
Log[2]])/((-75*x^2 + x^4 - x^5)*Log[2] + (75*x^2 - x^4 + x^5)*Log[(75 - x^ 
2 + x^3)/x]),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(31)=62\).

Time = 5.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06

method	result	size
parallelrisch	\(-\frac {-\ln \left (\frac {\ln \left (\frac {x^{3}-x^{2}+75}{x}\right )-\ln \left (2\right )}{\ln \left (2\right )}\right ) x -\ln \left (\frac {\ln \left (\frac {x^{3}-x^{2}+75}{x}\right )-\ln \left (2\right )}{\ln \left (2\right )}\right )}{x}\)	\(64\)

Input:

int((((-x^3+x^2-75)*ln((x^3-x^2+75)/x)+(x^3-x^2+75)*ln(2))*ln((ln((x^3-x^2 
+75)/x)-ln(2))/ln(2))+2*x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2)*ln((x^3-x^2 
+75)/x)+(-x^5+x^4-75*x^2)*ln(2)),x,method=_RETURNVERBOSE)
 

Output:

-1/x*(-ln((ln((x^3-x^2+75)/x)-ln(2))/ln(2))*x-ln((ln((x^3-x^2+75)/x)-ln(2) 
)/ln(2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\frac {{\left (x + 1\right )} \log \left (-\frac {\log \left (2\right ) - \log \left (\frac {x^{3} - x^{2} + 75}{x}\right )}{\log \left (2\right )}\right )}{x} \] Input:

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((lo 
g((x^3-x^2+75)/x)-log(2))/log(2))+2*x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2) 
*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="fricas")
 

Output:

(x + 1)*log(-(log(2) - log((x^3 - x^2 + 75)/x))/log(2))/x
 

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\log {\left (\log {\left (\frac {x^{3} - x^{2} + 75}{x} \right )} - \log {\left (2 \right )} \right )} + \frac {\log {\left (\frac {\log {\left (\frac {x^{3} - x^{2} + 75}{x} \right )} - \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )}}{x} \] Input:

integrate((((-x**3+x**2-75)*ln((x**3-x**2+75)/x)+(x**3-x**2+75)*ln(2))*ln( 
(ln((x**3-x**2+75)/x)-ln(2))/ln(2))+2*x**4+x**3-x**2-75*x-75)/((x**5-x**4+ 
75*x**2)*ln((x**3-x**2+75)/x)+(-x**5+x**4-75*x**2)*ln(2)),x)
 

Output:

log(log((x**3 - x**2 + 75)/x) - log(2)) + log((log((x**3 - x**2 + 75)/x) - 
 log(2))/log(2))/x
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\frac {{\left (x + 1\right )} \log \left (-\log \left (2\right ) + \log \left (x^{3} - x^{2} + 75\right ) - \log \left (x\right )\right ) - \log \left (\log \left (2\right )\right )}{x} \] Input:

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((lo 
g((x^3-x^2+75)/x)-log(2))/log(2))+2*x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2) 
*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="maxima")
 

Output:

((x + 1)*log(-log(2) + log(x^3 - x^2 + 75) - log(x)) - log(log(2)))/x
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\frac {\log \left (-\log \left (2\right ) + \log \left (x^{3} - x^{2} + 75\right ) - \log \left (x\right )\right )}{x} - \frac {\log \left (\log \left (2\right )\right )}{x} + \log \left (-\log \left (2\right ) + \log \left (x^{3} - x^{2} + 75\right ) - \log \left (x\right )\right ) \] Input:

integrate((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((lo 
g((x^3-x^2+75)/x)-log(2))/log(2))+2*x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2) 
*log((x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x, algorithm="giac")
 

Output:

log(-log(2) + log(x^3 - x^2 + 75) - log(x))/x - log(log(2))/x + log(-log(2 
) + log(x^3 - x^2 + 75) - log(x))
 

Mupad [B] (verification not implemented)

Time = 3.96 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\ln \left (\ln \left (\frac {x^3-x^2+75}{x}\right )-\ln \left (2\right )\right )-\frac {\ln \left (\ln \left (2\right )\right )}{x}+\frac {\ln \left (\ln \left (\frac {x^3-x^2+75}{x}\right )-\ln \left (2\right )\right )}{x} \] Input:

int(-(75*x - log((log((x^3 - x^2 + 75)/x) - log(2))/log(2))*(log(2)*(x^3 - 
 x^2 + 75) - log((x^3 - x^2 + 75)/x)*(x^3 - x^2 + 75)) + x^2 - x^3 - 2*x^4 
 + 75)/(log((x^3 - x^2 + 75)/x)*(75*x^2 - x^4 + x^5) - log(2)*(75*x^2 - x^ 
4 + x^5)),x)
 

Output:

log(log((x^3 - x^2 + 75)/x) - log(2)) - log(log(2))/x + log(log((x^3 - x^2 
 + 75)/x) - log(2))/x
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-75-75 x-x^2+x^3+2 x^4+\left (\left (75-x^2+x^3\right ) \log (2)+\left (-75+x^2-x^3\right ) \log \left (\frac {75-x^2+x^3}{x}\right )\right ) \log \left (\frac {-\log (2)+\log \left (\frac {75-x^2+x^3}{x}\right )}{\log (2)}\right )}{\left (-75 x^2+x^4-x^5\right ) \log (2)+\left (75 x^2-x^4+x^5\right ) \log \left (\frac {75-x^2+x^3}{x}\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {x^{3}-x^{2}+75}{x}\right )-\mathrm {log}\left (2\right )\right ) x +\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {x^{3}-x^{2}+75}{x}\right )-\mathrm {log}\left (2\right )}{\mathrm {log}\left (2\right )}\right )}{x} \] Input:

int((((-x^3+x^2-75)*log((x^3-x^2+75)/x)+(x^3-x^2+75)*log(2))*log((log((x^3 
-x^2+75)/x)-log(2))/log(2))+2*x^4+x^3-x^2-75*x-75)/((x^5-x^4+75*x^2)*log(( 
x^3-x^2+75)/x)+(-x^5+x^4-75*x^2)*log(2)),x)
 

Output:

(log(log((x**3 - x**2 + 75)/x) - log(2))*x + log((log((x**3 - x**2 + 75)/x 
) - log(2))/log(2)))/x