Integrand size = 123, antiderivative size = 29 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\log \left (e^{x/4}+\frac {x}{\log \left (\frac {5}{x^2}\right )}+\left (-1+\log \left (\frac {x}{3}\right )\right )^2\right ) \] Output:
ln((ln(1/3*x)-1)^2+x/ln(5/x^2)+exp(1/4*x))
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=-\log \left (\log \left (\frac {5}{x^2}\right )\right )+\log \left (4 \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )\right ) \] Input:
Integrate[(8*x + 4*x*Log[5/x^2] + (-8 + E^(x/4)*x)*Log[5/x^2]^2 + 8*Log[5/ x^2]^2*Log[x/3])/(4*x^2*Log[5/x^2] + (4*x + 4*E^(x/4)*x)*Log[5/x^2]^2 - 8* x*Log[5/x^2]^2*Log[x/3] + 4*x*Log[5/x^2]^2*Log[x/3]^2),x]
Output:
-Log[Log[5/x^2]] + Log[4*(x + Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^ 2 - 2*Log[x]))]
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.
| \(\displaystyle \int \frac {\left (e^{x/4} x-8\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log \left (\frac {x}{3}\right ) \log ^2\left (\frac {5}{x^2}\right )+4 x \log \left (\frac {5}{x^2}\right )+8 x}{4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+\left (4 e^{x/4} x+4 x\right ) \log ^2\left (\frac {5}{x^2}\right )+4 x^2 \log \left (\frac {5}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (e^{x/4} x-8\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log \left (\frac {x}{3}\right ) \log ^2\left (\frac {5}{x^2}\right )+4 x \log \left (\frac {5}{x^2}\right )+8 x}{4 x \log \left (\frac {5}{x^2}\right ) \left (\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)+x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {-\left (\left (8-e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )\right )+8 \log \left (\frac {x}{3}\right ) \log ^2\left (\frac {5}{x^2}\right )+4 x \log \left (\frac {5}{x^2}\right )+8 x}{x \log \left (\frac {5}{x^2}\right ) \left (\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {-\log \left (\frac {5}{x^2}\right ) x^2-(1+\log (9)) \log ^2\left (\frac {5}{x^2}\right ) x-\log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right ) x+4 \log \left (\frac {5}{x^2}\right ) x+2 \log ^2\left (\frac {5}{x^2}\right ) \log (x) x+8 x-8 \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{x \log \left (\frac {5}{x^2}\right ) \left (\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (4 \int \frac {1}{\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)}dx+8 \int \frac {1}{\log \left (\frac {5}{x^2}\right ) \left (\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}dx+8 \int \frac {\log \left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{x \left (\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}dx+2 \int \frac {\log \left (\frac {5}{x^2}\right ) \log (x)}{\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)}dx+\int \frac {x}{-\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-x-e^{x/4} \log \left (\frac {5}{x^2}\right )-(1+\log (9)) \log \left (\frac {5}{x^2}\right )+2 \log \left (\frac {5}{x^2}\right ) \log (x)}dx+(1+\log (9)) \int \frac {\log \left (\frac {5}{x^2}\right )}{-\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-x-e^{x/4} \log \left (\frac {5}{x^2}\right )-(1+\log (9)) \log \left (\frac {5}{x^2}\right )+2 \log \left (\frac {5}{x^2}\right ) \log (x)}dx+8 \int \frac {\log \left (\frac {5}{x^2}\right )}{x \left (-\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-x-e^{x/4} \log \left (\frac {5}{x^2}\right )-(1+\log (9)) \log \left (\frac {5}{x^2}\right )+2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}dx+\int \frac {\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )}{-\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-x-e^{x/4} \log \left (\frac {5}{x^2}\right )-(1+\log (9)) \log \left (\frac {5}{x^2}\right )+2 \log \left (\frac {5}{x^2}\right ) \log (x)}dx+x\right )\) |
Input:
Int[(8*x + 4*x*Log[5/x^2] + (-8 + E^(x/4)*x)*Log[5/x^2]^2 + 8*Log[5/x^2]^2 *Log[x/3])/(4*x^2*Log[5/x^2] + (4*x + 4*E^(x/4)*x)*Log[5/x^2]^2 - 8*x*Log[ 5/x^2]^2*Log[x/3] + 4*x*Log[5/x^2]^2*Log[x/3]^2),x]
Output:
$Aborted
Timed out.
\[\int \frac {8 \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (x \,{\mathrm e}^{\frac {x}{4}}-8\right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x \ln \left (\frac {5}{x^{2}}\right )+8 x}{4 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )^{2}-8 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (4 x \,{\mathrm e}^{\frac {x}{4}}+4 x \right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x^{2} \ln \left (\frac {5}{x^{2}}\right )}d x\]
Input:
int((8*ln(5/x^2)^2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x^2)^2+4*x*ln(5/x^2)+8* x)/(4*x*ln(5/x^2)^2*ln(1/3*x)^2-8*x*ln(5/x^2)^2*ln(1/3*x)+(4*x*exp(1/4*x)+ 4*x)*ln(5/x^2)^2+4*x^2*ln(5/x^2)),x)
Output:
int((8*ln(5/x^2)^2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x^2)^2+4*x*ln(5/x^2)+8* x)/(4*x*ln(5/x^2)^2*ln(1/3*x)^2-8*x*ln(5/x^2)^2*ln(1/3*x)+(4*x*exp(1/4*x)+ 4*x)*ln(5/x^2)^2+4*x^2*ln(5/x^2)),x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\log \left (-2 \, {\left (\log \left (\frac {5}{9}\right ) - 2\right )} \log \left (\frac {5}{x^{2}}\right )^{2} + \log \left (\frac {5}{x^{2}}\right )^{3} + {\left (\log \left (\frac {5}{9}\right )^{2} + 4 \, e^{\left (\frac {1}{4} \, x\right )} - 4 \, \log \left (\frac {5}{9}\right ) + 4\right )} \log \left (\frac {5}{x^{2}}\right ) + 4 \, x\right ) - \log \left (\log \left (\frac {5}{x^{2}}\right )\right ) \] Input:
integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log (5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4 *x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="fricas")
Output:
log(-2*(log(5/9) - 2)*log(5/x^2)^2 + log(5/x^2)^3 + (log(5/9)^2 + 4*e^(1/4 *x) - 4*log(5/9) + 4)*log(5/x^2) + 4*x) - log(log(5/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (22) = 44\).
Time = 2.84 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.34 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\log {\left (e^{\frac {x}{4}} + \frac {- x + 2 \log {\left (\frac {x}{3} \right )}^{3} - 4 \log {\left (\frac {x}{3} \right )}^{2} - \log {\left (5 \right )} \log {\left (\frac {x}{3} \right )}^{2} + 2 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )}^{2} - 4 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} + 2 \log {\left (\frac {x}{3} \right )} + 2 \log {\left (5 \right )} \log {\left (\frac {x}{3} \right )} - \log {\left (5 \right )} + 2 \log {\left (3 \right )}}{2 \log {\left (\frac {x}{3} \right )} - \log {\left (5 \right )} + 2 \log {\left (3 \right )}} \right )} \] Input:
integrate((8*ln(5/x**2)**2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x**2)**2+4*x*ln (5/x**2)+8*x)/(4*x*ln(5/x**2)**2*ln(1/3*x)**2-8*x*ln(5/x**2)**2*ln(1/3*x)+ (4*x*exp(1/4*x)+4*x)*ln(5/x**2)**2+4*x**2*ln(5/x**2)),x)
Output:
log(exp(x/4) + (-x + 2*log(x/3)**3 - 4*log(x/3)**2 - log(5)*log(x/3)**2 + 2*log(3)*log(x/3)**2 - 4*log(3)*log(x/3) + 2*log(x/3) + 2*log(5)*log(x/3) - log(5) + 2*log(3))/(2*log(x/3) - log(5) + 2*log(3)))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\log \left (\frac {\log \left (5\right ) \log \left (3\right )^{2} + {\left (\log \left (5\right ) + 4 \, \log \left (3\right ) + 4\right )} \log \left (x\right )^{2} - 2 \, \log \left (x\right )^{3} + {\left (\log \left (5\right ) - 2 \, \log \left (x\right )\right )} e^{\left (\frac {1}{4} \, x\right )} + 2 \, \log \left (5\right ) \log \left (3\right ) - 2 \, {\left ({\left (\log \left (5\right ) + 2\right )} \log \left (3\right ) + \log \left (3\right )^{2} + \log \left (5\right ) + 1\right )} \log \left (x\right ) + x + \log \left (5\right )}{\log \left (5\right ) - 2 \, \log \left (x\right )}\right ) \] Input:
integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log (5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4 *x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="maxima")
Output:
log((log(5)*log(3)^2 + (log(5) + 4*log(3) + 4)*log(x)^2 - 2*log(x)^3 + (lo g(5) - 2*log(x))*e^(1/4*x) + 2*log(5)*log(3) - 2*((log(5) + 2)*log(3) + lo g(3)^2 + log(5) + 1)*log(x) + x + log(5))/(log(5) - 2*log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.55 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\log \left (\log \left (5\right ) \log \left (3\right )^{2} - 2 \, \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, \log \left (3\right )^{2} \log \left (x\right ) + \log \left (5\right ) \log \left (x\right )^{2} + 4 \, \log \left (3\right ) \log \left (x\right )^{2} - 2 \, \log \left (x\right )^{3} + e^{\left (\frac {1}{4} \, x\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (3\right ) - 2 \, e^{\left (\frac {1}{4} \, x\right )} \log \left (x\right ) - 2 \, \log \left (5\right ) \log \left (x\right ) - 4 \, \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + x + \log \left (5\right ) - 2 \, \log \left (x\right )\right ) - \log \left (\log \left (5\right ) - 2 \, \log \left (x\right )\right ) \] Input:
integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log (5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4 *x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="giac")
Output:
log(log(5)*log(3)^2 - 2*log(5)*log(3)*log(x) - 2*log(3)^2*log(x) + log(5)* log(x)^2 + 4*log(3)*log(x)^2 - 2*log(x)^3 + e^(1/4*x)*log(5) + 2*log(5)*lo g(3) - 2*e^(1/4*x)*log(x) - 2*log(5)*log(x) - 4*log(3)*log(x) + 4*log(x)^2 + x + log(5) - 2*log(x)) - log(log(5) - 2*log(x))
Timed out. \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=\int \frac {8\,x+4\,x\,\ln \left (\frac {5}{x^2}\right )+8\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+{\ln \left (\frac {5}{x^2}\right )}^2\,\left (x\,{\mathrm {e}}^{x/4}-8\right )}{{\ln \left (\frac {5}{x^2}\right )}^2\,\left (4\,x+4\,x\,{\mathrm {e}}^{x/4}\right )+4\,x^2\,\ln \left (\frac {5}{x^2}\right )-8\,x\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+4\,x\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \left (\frac {5}{x^2}\right )}^2} \,d x \] Input:
int((8*x + 4*x*log(5/x^2) + 8*log(x/3)*log(5/x^2)^2 + log(5/x^2)^2*(x*exp( x/4) - 8))/(log(5/x^2)^2*(4*x + 4*x*exp(x/4)) + 4*x^2*log(5/x^2) - 8*x*log (x/3)*log(5/x^2)^2 + 4*x*log(x/3)^2*log(5/x^2)^2),x)
Output:
int((8*x + 4*x*log(5/x^2) + 8*log(x/3)*log(5/x^2)^2 + log(5/x^2)^2*(x*exp( x/4) - 8))/(log(5/x^2)^2*(4*x + 4*x*exp(x/4)) + 4*x^2*log(5/x^2) - 8*x*log (x/3)*log(5/x^2)^2 + 4*x*log(x/3)^2*log(5/x^2)^2), x)
Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\frac {5}{x^{2}}\right )\right )+\mathrm {log}\left (e^{\frac {x}{4}} \mathrm {log}\left (\frac {5}{x^{2}}\right )+\mathrm {log}\left (\frac {x}{3}\right )^{2} \mathrm {log}\left (\frac {5}{x^{2}}\right )-2 \,\mathrm {log}\left (\frac {x}{3}\right ) \mathrm {log}\left (\frac {5}{x^{2}}\right )+\mathrm {log}\left (\frac {5}{x^{2}}\right )+x \right ) \] Input:
int((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log(5/x^2 )+8*x)/(4*x*log(5/x^2)^2*log(1/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4*x*exp (1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x)
Output:
- log(log(5/x**2)) + log(e**(x/4)*log(5/x**2) + log(x/3)**2*log(5/x**2) - 2*log(x/3)*log(5/x**2) + log(5/x**2) + x)