\(\int \frac {(-7+5 x^2) \log (\frac {1}{5} (7-5 x^2))+(10 x^2+(7-5 x^2) \log (\frac {1}{5} (7-5 x^2))+\log (\frac {5}{x}) (10 x^2+(7-5 x^2) \log (\frac {1}{5} (7-5 x^2)))) \log (1+\log (\frac {5}{x}))+(-7 x+5 x^3+(-7 x+5 x^3) \log (\frac {5}{x})) \log ^2(1+\log (\frac {5}{x}))}{(-7 x^2+5 x^4+(-7 x^2+5 x^4) \log (\frac {5}{x})) \log ^2(1+\log (\frac {5}{x}))} \, dx\) [2957]

Optimal result

Integrand size = 170, antiderivative size = 28 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \] Output:

ln(-x^2+7/5)/ln(ln(5/x)+1)/x+ln(x)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \] Input:

Integrate[((-7 + 5*x^2)*Log[(7 - 5*x^2)/5] + (10*x^2 + (7 - 5*x^2)*Log[(7 
- 5*x^2)/5] + Log[5/x]*(10*x^2 + (7 - 5*x^2)*Log[(7 - 5*x^2)/5]))*Log[1 + 
Log[5/x]] + (-7*x + 5*x^3 + (-7*x + 5*x^3)*Log[5/x])*Log[1 + Log[5/x]]^2)/ 
((-7*x^2 + 5*x^4 + (-7*x^2 + 5*x^4)*Log[5/x])*Log[1 + Log[5/x]]^2),x]
 

Output:

Log[x] + Log[7/5 - x^2]/(x*Log[1 + Log[5/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.

\(\displaystyle \int \frac {\left (5 x^3+\left (5 x^3-7 x\right ) \log \left (\frac {5}{x}\right )-7 x\right ) \log ^2\left (\log \left (\frac {5}{x}\right )+1\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )+1\right )+\left (5 x^2-7\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )}{\left (5 x^4-7 x^2+\left (5 x^4-7 x^2\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (\log \left (\frac {5}{x}\right )+1\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\log \left (\frac {7}{5}-x^2\right ) \left (\frac {1}{\log \left (\frac {5}{x}\right )+1}-\log \left (\log \left (\frac {5}{x}\right )+1\right )\right )+x \log \left (\log \left (\frac {5}{x}\right )+1\right ) \left (\frac {10 x}{5 x^2-7}+\log \left (\log \left (\frac {5}{x}\right )+1\right )\right )}{x^2 \log ^2\left (\log \left (\frac {5}{x}\right )+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (\log \left (\frac {5}{x}\right )+1\right ) \log ^2\left (\log \left (\frac {5}{x}\right )+1\right )}+\frac {10 x^2-5 x^2 \log \left (\frac {7}{5}-x^2\right )+7 \log \left (\frac {7}{5}-x^2\right )}{x^2 \left (5 x^2-7\right ) \log \left (\log \left (\frac {5}{x}\right )+1\right )}+\frac {1}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (\log \left (\frac {5}{x}\right )+1\right ) \log ^2\left (\log \left (\frac {5}{x}\right )+1\right )}dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (\log \left (\frac {5}{x}\right )+1\right )}dx-\frac {5 \int \frac {1}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (\log \left (\frac {5}{x}\right )+1\right )}dx}{\sqrt {7}}-\frac {5 \int \frac {1}{\left (\sqrt {5} x+\sqrt {7}\right ) \log \left (\log \left (\frac {5}{x}\right )+1\right )}dx}{\sqrt {7}}+\log (x)\)

Input:

Int[((-7 + 5*x^2)*Log[(7 - 5*x^2)/5] + (10*x^2 + (7 - 5*x^2)*Log[(7 - 5*x^ 
2)/5] + Log[5/x]*(10*x^2 + (7 - 5*x^2)*Log[(7 - 5*x^2)/5]))*Log[1 + Log[5/ 
x]] + (-7*x + 5*x^3 + (-7*x + 5*x^3)*Log[5/x])*Log[1 + Log[5/x]]^2)/((-7*x 
^2 + 5*x^4 + (-7*x^2 + 5*x^4)*Log[5/x])*Log[1 + Log[5/x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 55.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
risch	\(\ln \left (x \right )+\frac {\ln \left (-x^{2}+\frac {7}{5}\right )}{x \ln \left (\ln \left (5\right )-\ln \left (x \right )+1\right )}\)	\(27\)
parallelrisch	\(\frac {210 \ln \left (x \right ) x \ln \left (\ln \left (\frac {5}{x}\right )+1\right )+210 \ln \left (-x^{2}+\frac {7}{5}\right )}{210 x \ln \left (\ln \left (\frac {5}{x}\right )+1\right )}\)	\(42\)

Input:

int((((5*x^3-7*x)*ln(5/x)+5*x^3-7*x)*ln(ln(5/x)+1)^2+(((-5*x^2+7)*ln(-x^2+ 
7/5)+10*x^2)*ln(5/x)+(-5*x^2+7)*ln(-x^2+7/5)+10*x^2)*ln(ln(5/x)+1)+(5*x^2- 
7)*ln(-x^2+7/5))/((5*x^4-7*x^2)*ln(5/x)+5*x^4-7*x^2)/ln(ln(5/x)+1)^2,x,met 
hod=_RETURNVERBOSE)
 

Output:

ln(x)+ln(-x^2+7/5)/x/ln(ln(5)-ln(x)+1)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=-\frac {x \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right ) + 1\right ) - \log \left (-x^{2} + \frac {7}{5}\right )}{x \log \left (\log \left (\frac {5}{x}\right ) + 1\right )} \] Input:

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7) 
*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5 
/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log(l 
og(5/x)+1)^2,x, algorithm="fricas")
 

Output:

-(x*log(5/x)*log(log(5/x) + 1) - log(-x^2 + 7/5))/(x*log(log(5/x) + 1))
 

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\log {\left (x \right )} + \frac {\log {\left (\frac {7}{5} - x^{2} \right )}}{x \log {\left (\log {\left (\frac {5}{x} \right )} + 1 \right )}} \] Input:

integrate((((5*x**3-7*x)*ln(5/x)+5*x**3-7*x)*ln(ln(5/x)+1)**2+(((-5*x**2+7 
)*ln(-x**2+7/5)+10*x**2)*ln(5/x)+(-5*x**2+7)*ln(-x**2+7/5)+10*x**2)*ln(ln( 
5/x)+1)+(5*x**2-7)*ln(-x**2+7/5))/((5*x**4-7*x**2)*ln(5/x)+5*x**4-7*x**2)/ 
ln(ln(5/x)+1)**2,x)
 

Output:

log(x) + log(7/5 - x**2)/(x*log(log(5/x) + 1))
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=-\frac {\log \left (5\right ) - \log \left (-5 \, x^{2} + 7\right )}{x \log \left (\log \left (5\right ) - \log \left (x\right ) + 1\right )} + \log \left (x\right ) \] Input:

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7) 
*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5 
/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log(l 
og(5/x)+1)^2,x, algorithm="maxima")
 

Output:

-(log(5) - log(-5*x^2 + 7))/(x*log(log(5) - log(x) + 1)) + log(x)
 

Giac [F]

\[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\int { \frac {{\left (5 \, x^{3} + {\left (5 \, x^{3} - 7 \, x\right )} \log \left (\frac {5}{x}\right ) - 7 \, x\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2} + {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right )\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )}{{\left (5 \, x^{4} - 7 \, x^{2} + {\left (5 \, x^{4} - 7 \, x^{2}\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2}} \,d x } \] Input:

integrate((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7) 
*log(-x^2+7/5)+10*x^2)*log(5/x)+(-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5 
/x)+1)+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log(l 
og(5/x)+1)^2,x, algorithm="giac")
 

Output:

integrate(((5*x^3 + (5*x^3 - 7*x)*log(5/x) - 7*x)*log(log(5/x) + 1)^2 + (5 
*x^2 - 7)*log(-x^2 + 7/5) + (10*x^2 - (5*x^2 - 7)*log(-x^2 + 7/5) + (10*x^ 
2 - (5*x^2 - 7)*log(-x^2 + 7/5))*log(5/x))*log(log(5/x) + 1))/((5*x^4 - 7* 
x^2 + (5*x^4 - 7*x^2)*log(5/x))*log(log(5/x) + 1)^2), x)
 

Mupad [B] (verification not implemented)

Time = 4.46 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.96 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\ln \left (x\right )+\ln \left (\frac {5}{x}\right )\,\left (\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}\right )+\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}+\frac {\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}+\frac {\ln \left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (7\,\ln \left (\frac {7}{5}-x^2\right )-5\,x^2\,\ln \left (\frac {7}{5}-x^2\right )+10\,x^2\right )}{x\,\left (5\,x^2-7\right )}}{\ln \left (\ln \left (\frac {5}{x}\right )+1\right )} \] Input:

int(-(log(log(5/x) + 1)*(log(5/x)*(10*x^2 - log(7/5 - x^2)*(5*x^2 - 7)) + 
10*x^2 - log(7/5 - x^2)*(5*x^2 - 7)) - log(log(5/x) + 1)^2*(7*x + log(5/x) 
*(7*x - 5*x^3) - 5*x^3) + log(7/5 - x^2)*(5*x^2 - 7))/(log(log(5/x) + 1)^2 
*(log(5/x)*(7*x^2 - 5*x^4) + 7*x^2 - 5*x^4)),x)
 

Output:

log(x) + log(5/x)*(log(7/5 - x^2)/x - (2*x)/(x^2 - 7/5)) + log(7/5 - x^2)/ 
x - (2*x)/(x^2 - 7/5) + (log(7/5 - x^2)/x + (log(log(5/x) + 1)*(log(5/x) + 
 1)*(7*log(7/5 - x^2) - 5*x^2*log(7/5 - x^2) + 10*x^2))/(x*(5*x^2 - 7)))/l 
og(log(5/x) + 1)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {5}{x}\right )+1\right ) \mathrm {log}\left (x \right ) x +\mathrm {log}\left (-x^{2}+\frac {7}{5}\right )}{\mathrm {log}\left (\mathrm {log}\left (\frac {5}{x}\right )+1\right ) x} \] Input:

int((((5*x^3-7*x)*log(5/x)+5*x^3-7*x)*log(log(5/x)+1)^2+(((-5*x^2+7)*log(- 
x^2+7/5)+10*x^2)*log(5/x)+(-5*x^2+7)*log(-x^2+7/5)+10*x^2)*log(log(5/x)+1) 
+(5*x^2-7)*log(-x^2+7/5))/((5*x^4-7*x^2)*log(5/x)+5*x^4-7*x^2)/log(log(5/x 
)+1)^2,x)
 

Output:

(log(log(5/x) + 1)*log(x)*x + log(( - 5*x**2 + 7)/5))/(log(log(5/x) + 1)*x 
)