Integrand size = 78, antiderivative size = 22 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=1-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \] Output:
12/x^3/ln(ln(x^2)-x)+1-ln(x)
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \] Input:
Integrate[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x ^3*Log[x^2])*Log[-x + Log[x^2]]^2)/((-x^5 + x^4*Log[x^2])*Log[-x + Log[x^2 ]]^2),x]
Output:
-Log[x] + 12/(x^3*Log[-x + Log[x^2]])
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.
\(\displaystyle \int \frac {\left (36 x-36 \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )-x\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )+12 x-24}{\left (x^4 \log \left (x^2\right )-x^5\right ) \log ^2\left (\log \left (x^2\right )-x\right )} \, dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {\left (36 x-36 \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )-x\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )+12 x-24}{x^4 \left (\log \left (x^2\right )-x\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {12 (x-2)}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}-\frac {36}{x^4 \log \left (\log \left (x^2\right )-x\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 24 \int \frac {1}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx-36 \int \frac {1}{x^4 \log \left (\log \left (x^2\right )-x\right )}dx-12 \int \frac {1}{x^3 \left (x-\log \left (x^2\right )\right ) \log ^2\left (\log \left (x^2\right )-x\right )}dx-\log (x)\) |
Input:
Int[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x^3*Log [x^2])*Log[-x + Log[x^2]]^2)/((-x^5 + x^4*Log[x^2])*Log[-x + Log[x^2]]^2), x]
Output:
$Aborted
Time = 0.44 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68
method | result | size |
parallelrisch | \(\frac {24-x^{3} \ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )-x \right )}{2 x^{3} \ln \left (\ln \left (x^{2}\right )-x \right )}\) | \(37\) |
Input:
int(((-x^3*ln(x^2)+x^4)*ln(ln(x^2)-x)^2+(-36*ln(x^2)+36*x)*ln(ln(x^2)-x)+1 2*x-24)/(x^4*ln(x^2)-x^5)/ln(ln(x^2)-x)^2,x,method=_RETURNVERBOSE)
Output:
1/2*(24-x^3*ln(x^2)*ln(ln(x^2)-x))/x^3/ln(ln(x^2)-x)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\frac {x^{3} \log \left (x^{2}\right ) \log \left (-x + \log \left (x^{2}\right )\right ) - 24}{2 \, x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} \] Input:
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm="fri cas")
Output:
-1/2*(x^3*log(x^2)*log(-x + log(x^2)) - 24)/(x^3*log(-x + log(x^2)))
Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=- \log {\left (x \right )} + \frac {12}{x^{3} \log {\left (- x + \log {\left (x^{2} \right )} \right )}} \] Input:
integrate(((-x**3*ln(x**2)+x**4)*ln(ln(x**2)-x)**2+(-36*ln(x**2)+36*x)*ln( ln(x**2)-x)+12*x-24)/(x**4*ln(x**2)-x**5)/ln(ln(x**2)-x)**2,x)
Output:
-log(x) + 12/(x**3*log(-x + log(x**2)))
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + 2 \, \log \left (x\right )\right )} - \log \left (x\right ) \] Input:
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm="max ima")
Output:
12/(x^3*log(-x + 2*log(x))) - log(x)
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} - \log \left (x\right ) \] Input:
integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(l og(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x, algorithm="gia c")
Output:
12/(x^3*log(-x + log(x^2))) - log(x)
Time = 3.51 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {36}{2\,x^2-x^3}-\frac {36\,x}{2\,x^3-x^4}-\ln \left (x\right )+\frac {12}{x^3\,\ln \left (\ln \left (x^2\right )-x\right )} \] Input:
int((12*x + log(log(x^2) - x)*(36*x - 36*log(x^2)) - log(log(x^2) - x)^2*( x^3*log(x^2) - x^4) - 24)/(log(log(x^2) - x)^2*(x^4*log(x^2) - x^5)),x)
Output:
36/(2*x^2 - x^3) - (36*x)/(2*x^3 - x^4) - log(x) + 12/(x^3*log(log(x^2) - x))
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {-\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )-x \right ) \mathrm {log}\left (x \right ) x^{3}+12}{\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )-x \right ) x^{3}} \] Input:
int(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(log(x^2 )-x)+12*x-24)/(x^4*log(x^2)-x^5)/log(log(x^2)-x)^2,x)
Output:
( - log(log(x**2) - x)*log(x)*x**3 + 12)/(log(log(x**2) - x)*x**3)