Integrand size = 76, antiderivative size = 27 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\left (-\frac {5}{3}+x-x^2\right ) \left (4+\frac {x^3}{2 \log \left (\log \left (x^2\right )\right )}\right ) \] Output:
4*(1+1/8*x^3/ln(ln(x^2)))*(x-x^2-5/3)
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=4 x-4 x^2-\frac {x^3 \left (5-3 x+3 x^2\right )}{6 \log \left (\log \left (x^2\right )\right )} \] Input:
Integrate[(10*x^2 - 6*x^3 + 6*x^4 + (-15*x^2 + 12*x^3 - 15*x^4)*Log[x^2]*L og[Log[x^2]] + (24 - 48*x)*Log[x^2]*Log[Log[x^2]]^2)/(6*Log[x^2]*Log[Log[x ^2]]^2),x]
Output:
4*x - 4*x^2 - (x^3*(5 - 3*x + 3*x^2))/(6*Log[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.
\(\displaystyle \int \frac {6 x^4-6 x^3+10 x^2+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-15 x^4+12 x^3-15 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int \frac {6 x^4-6 x^3+10 x^2+24 (1-2 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )-3 \left (5 x^4-4 x^3+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{\log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{6} \int \left (-\frac {3 \left (5 x^2-4 x+5\right ) x^2}{\log \left (\log \left (x^2\right )\right )}+\frac {2 \left (3 x^2-3 x+5\right ) x^2}{\log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}-24 (2 x-1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (-3 \text {Subst}\left (\int \frac {x}{\log (x) \log ^2(\log (x))}dx,x,x^2\right )+6 \text {Subst}\left (\int \frac {x}{\log (\log (x))}dx,x,x^2\right )+10 \int \frac {x^2}{\log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}dx-15 \int \frac {x^2}{\log \left (\log \left (x^2\right )\right )}dx+6 \int \frac {x^4}{\log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}dx-15 \int \frac {x^4}{\log \left (\log \left (x^2\right )\right )}dx-6 (1-2 x)^2\right )\) |
Input:
Int[(10*x^2 - 6*x^3 + 6*x^4 + (-15*x^2 + 12*x^3 - 15*x^4)*Log[x^2]*Log[Log [x^2]] + (24 - 48*x)*Log[x^2]*Log[Log[x^2]]^2)/(6*Log[x^2]*Log[Log[x^2]]^2 ),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63
method | result | size |
parallelrisch | \(-\frac {3 x^{5}-3 x^{4}+5 x^{3}+24 \ln \left (\ln \left (x^{2}\right )\right ) x^{2}-24 \ln \left (\ln \left (x^{2}\right )\right ) x}{6 \ln \left (\ln \left (x^{2}\right )\right )}\) | \(44\) |
Input:
int(1/6*((-48*x+24)*ln(x^2)*ln(ln(x^2))^2+(-15*x^4+12*x^3-15*x^2)*ln(x^2)* ln(ln(x^2))+6*x^4-6*x^3+10*x^2)/ln(x^2)/ln(ln(x^2))^2,x,method=_RETURNVERB OSE)
Output:
-1/6*(3*x^5-3*x^4+5*x^3+24*ln(ln(x^2))*x^2-24*ln(ln(x^2))*x)/ln(ln(x^2))
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=-\frac {3 \, x^{5} - 3 \, x^{4} + 5 \, x^{3} + 24 \, {\left (x^{2} - x\right )} \log \left (\log \left (x^{2}\right )\right )}{6 \, \log \left (\log \left (x^{2}\right )\right )} \] Input:
integrate(1/6*((-48*x+24)*log(x^2)*log(log(x^2))^2+(-15*x^4+12*x^3-15*x^2) *log(x^2)*log(log(x^2))+6*x^4-6*x^3+10*x^2)/log(x^2)/log(log(x^2))^2,x, al gorithm="fricas")
Output:
-1/6*(3*x^5 - 3*x^4 + 5*x^3 + 24*(x^2 - x)*log(log(x^2)))/log(log(x^2))
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=- 4 x^{2} + 4 x + \frac {- 3 x^{5} + 3 x^{4} - 5 x^{3}}{6 \log {\left (\log {\left (x^{2} \right )} \right )}} \] Input:
integrate(1/6*((-48*x+24)*ln(x**2)*ln(ln(x**2))**2+(-15*x**4+12*x**3-15*x* *2)*ln(x**2)*ln(ln(x**2))+6*x**4-6*x**3+10*x**2)/ln(x**2)/ln(ln(x**2))**2, x)
Output:
-4*x**2 + 4*x + (-3*x**5 + 3*x**4 - 5*x**3)/(6*log(log(x**2)))
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=-4 \, x^{2} + 4 \, x - \frac {3 \, x^{5} - 3 \, x^{4} + 5 \, x^{3}}{6 \, {\left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right )}} \] Input:
integrate(1/6*((-48*x+24)*log(x^2)*log(log(x^2))^2+(-15*x^4+12*x^3-15*x^2) *log(x^2)*log(log(x^2))+6*x^4-6*x^3+10*x^2)/log(x^2)/log(log(x^2))^2,x, al gorithm="maxima")
Output:
-4*x^2 + 4*x - 1/6*(3*x^5 - 3*x^4 + 5*x^3)/(log(2) + log(log(x)))
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=-4 \, x^{2} + 4 \, x - \frac {3 \, x^{5} - 3 \, x^{4} + 5 \, x^{3}}{6 \, \log \left (\log \left (x^{2}\right )\right )} \] Input:
integrate(1/6*((-48*x+24)*log(x^2)*log(log(x^2))^2+(-15*x^4+12*x^3-15*x^2) *log(x^2)*log(log(x^2))+6*x^4-6*x^3+10*x^2)/log(x^2)/log(log(x^2))^2,x, al gorithm="giac")
Output:
-4*x^2 + 4*x - 1/6*(3*x^5 - 3*x^4 + 5*x^3)/log(log(x^2))
Time = 3.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.96 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=4\,x-\frac {\frac {x^3\,\left (3\,x^2-3\,x+5\right )}{6}-\frac {x^3\,\ln \left (x^2\right )\,\ln \left (\ln \left (x^2\right )\right )\,\left (5\,x^2-4\,x+5\right )}{4}}{\ln \left (\ln \left (x^2\right )\right )}-\ln \left (x^2\right )\,\left (\frac {5\,x^5}{4}-x^4+\frac {5\,x^3}{4}\right )-4\,x^2 \] Input:
int(-(x^3 - (5*x^2)/3 - x^4 + (log(x^2)*log(log(x^2))*(15*x^2 - 12*x^3 + 1 5*x^4))/6 + (log(x^2)*log(log(x^2))^2*(48*x - 24))/6)/(log(x^2)*log(log(x^ 2))^2),x)
Output:
4*x - ((x^3*(3*x^2 - 3*x + 5))/6 - (x^3*log(x^2)*log(log(x^2))*(5*x^2 - 4* x + 5))/4)/log(log(x^2)) - log(x^2)*((5*x^3)/4 - x^4 + (5*x^5)/4) - 4*x^2
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {10 x^2-6 x^3+6 x^4+\left (-15 x^2+12 x^3-15 x^4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(24-48 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{6 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\frac {x \left (-24 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right ) x +24 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )-3 x^{4}+3 x^{3}-5 x^{2}\right )}{6 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )} \] Input:
int(1/6*((-48*x+24)*log(x^2)*log(log(x^2))^2+(-15*x^4+12*x^3-15*x^2)*log(x ^2)*log(log(x^2))+6*x^4-6*x^3+10*x^2)/log(x^2)/log(log(x^2))^2,x)
Output:
(x*( - 24*log(log(x**2))*x + 24*log(log(x**2)) - 3*x**4 + 3*x**3 - 5*x**2) )/(6*log(log(x**2)))