Integrand size = 119, antiderivative size = 25 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 x+\frac {4 x}{\log \left (\frac {2 (3-x)}{x+\log \left (x^2\right )}\right )} \] Output:
4*x/ln(2*(3-x)/(ln(x^2)+x))-4*x
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 \left (x-\frac {x}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}\right ) \] Input:
Integrate[(-24 - 4*x - 4*x*Log[x^2] + (-12*x + 4*x^2 + (-12 + 4*x)*Log[x^2 ])*Log[(6 - 2*x)/(x + Log[x^2])] + (12*x - 4*x^2 + (12 - 4*x)*Log[x^2])*Lo g[(6 - 2*x)/(x + Log[x^2])]^2)/((-3*x + x^2 + (-3 + x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2),x]
Output:
-4*(x - x/Log[(6 - 2*x)/(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.
\(\displaystyle \int \frac {\left (-4 x^2+(12-4 x) \log \left (x^2\right )+12 x\right ) \log ^2\left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )+\left (4 x^2+(4 x-12) \log \left (x^2\right )-12 x\right ) \log \left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )-4 x \log \left (x^2\right )-4 x-24}{\left (x^2+(x-3) \log \left (x^2\right )-3 x\right ) \log ^2\left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (\left (-4 x^2+(12-4 x) \log \left (x^2\right )+12 x\right ) \log ^2\left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )\right )-\left (4 x^2+(4 x-12) \log \left (x^2\right )-12 x\right ) \log \left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )+4 x \log \left (x^2\right )+4 x+24}{(3-x) \left (\log \left (x^2\right )+x\right ) \log ^2\left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 \left (x \log \left (x^2\right )+x+6\right )}{(x-3) \left (\log \left (x^2\right )+x\right ) \log ^2\left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )}+\frac {4}{\log \left (\frac {6-2 x}{\log \left (x^2\right )+x}\right )}-4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}dx-36 \int \frac {1}{(x-3) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}dx-4 \int \frac {\log \left (x^2\right )}{\left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}dx-12 \int \frac {\log \left (x^2\right )}{(x-3) \left (x+\log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}dx+4 \int \frac {1}{\log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}dx-4 x\) |
Input:
Int[(-24 - 4*x - 4*x*Log[x^2] + (-12*x + 4*x^2 + (-12 + 4*x)*Log[x^2])*Log [(6 - 2*x)/(x + Log[x^2])] + (12*x - 4*x^2 + (12 - 4*x)*Log[x^2])*Log[(6 - 2*x)/(x + Log[x^2])]^2)/((-3*x + x^2 + (-3 + x)*Log[x^2])*Log[(6 - 2*x)/( x + Log[x^2])]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
Time = 0.91 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24
method | result | size |
parallelrisch | \(\frac {-8 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right ) x +8 x -48 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right )}{2 \ln \left (-\frac {2 \left (-3+x \right )}{\ln \left (x^{2}\right )+x}\right )}\) | \(56\) |
Input:
int((((-4*x+12)*ln(x^2)-4*x^2+12*x)*ln((6-2*x)/(ln(x^2)+x))^2+((4*x-12)*ln (x^2)+4*x^2-12*x)*ln((6-2*x)/(ln(x^2)+x))-4*x*ln(x^2)-4*x-24)/((-3+x)*ln(x ^2)+x^2-3*x)/ln((6-2*x)/(ln(x^2)+x))^2,x,method=_RETURNVERBOSE)
Output:
1/2*(-8*ln(-2*(-3+x)/(ln(x^2)+x))*x+8*x-48*ln(-2*(-3+x)/(ln(x^2)+x)))/ln(- 2*(-3+x)/(ln(x^2)+x))
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-\frac {4 \, {\left (x \log \left (-\frac {2 \, {\left (x - 3\right )}}{x + \log \left (x^{2}\right )}\right ) - x\right )}}{\log \left (-\frac {2 \, {\left (x - 3\right )}}{x + \log \left (x^{2}\right )}\right )} \] Input:
integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4 *x-12)*log(x^2)+4*x^2-12*x)*log((6-2*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24) /((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="frica s")
Output:
-4*(x*log(-2*(x - 3)/(x + log(x^2))) - x)/log(-2*(x - 3)/(x + log(x^2)))
Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=- 4 x + \frac {4 x}{\log {\left (\frac {6 - 2 x}{x + \log {\left (x^{2} \right )}} \right )}} \] Input:
integrate((((-4*x+12)*ln(x**2)-4*x**2+12*x)*ln((6-2*x)/(ln(x**2)+x))**2+(( 4*x-12)*ln(x**2)+4*x**2-12*x)*ln((6-2*x)/(ln(x**2)+x))-4*x*ln(x**2)-4*x-24 )/((-3+x)*ln(x**2)+x**2-3*x)/ln((6-2*x)/(ln(x**2)+x))**2,x)
Output:
-4*x + 4*x/log((6 - 2*x)/(x + log(x**2)))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-\frac {4 \, {\left ({\left (-i \, \pi - \log \left (2\right ) + 1\right )} x + x \log \left (x + 2 \, \log \left (x\right )\right ) - x \log \left (x - 3\right )\right )}}{-i \, \pi - \log \left (2\right ) + \log \left (x + 2 \, \log \left (x\right )\right ) - \log \left (x - 3\right )} \] Input:
integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4 *x-12)*log(x^2)+4*x^2-12*x)*log((6-2*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24) /((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="maxim a")
Output:
-4*((-I*pi - log(2) + 1)*x + x*log(x + 2*log(x)) - x*log(x - 3))/(-I*pi - log(2) + log(x + 2*log(x)) - log(x - 3))
Time = 0.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=-4 \, x - \frac {4 \, x}{\log \left (x + \log \left (x^{2}\right )\right ) - \log \left (-2 \, x + 6\right )} \] Input:
integrate((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4 *x-12)*log(x^2)+4*x^2-12*x)*log((6-2*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24) /((-3+x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x, algorithm="giac" )
Output:
-4*x - 4*x/(log(x + log(x^2)) - log(-2*x + 6))
Time = 0.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=\frac {4\,x}{\ln \left (-\frac {2\,x-6}{x+\ln \left (x^2\right )}\right )}-4\,x \] Input:
int(-(4*x + 4*x*log(x^2) - log(-(2*x - 6)/(x + log(x^2)))*(4*x^2 - 12*x + log(x^2)*(4*x - 12)) + log(-(2*x - 6)/(x + log(x^2)))^2*(4*x^2 - 12*x + lo g(x^2)*(4*x - 12)) + 24)/(log(-(2*x - 6)/(x + log(x^2)))^2*(log(x^2)*(x - 3) - 3*x + x^2)),x)
Output:
(4*x)/log(-(2*x - 6)/(x + log(x^2))) - 4*x
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {-24-4 x-4 x \log \left (x^2\right )+\left (-12 x+4 x^2+(-12+4 x) \log \left (x^2\right )\right ) \log \left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )+\left (12 x-4 x^2+(12-4 x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )}{\left (-3 x+x^2+(-3+x) \log \left (x^2\right )\right ) \log ^2\left (\frac {6-2 x}{x+\log \left (x^2\right )}\right )} \, dx=\frac {4 x \left (-\mathrm {log}\left (\frac {-2 x +6}{\mathrm {log}\left (x^{2}\right )+x}\right )+1\right )}{\mathrm {log}\left (\frac {-2 x +6}{\mathrm {log}\left (x^{2}\right )+x}\right )} \] Input:
int((((-4*x+12)*log(x^2)-4*x^2+12*x)*log((6-2*x)/(log(x^2)+x))^2+((4*x-12) *log(x^2)+4*x^2-12*x)*log((6-2*x)/(log(x^2)+x))-4*x*log(x^2)-4*x-24)/((-3+ x)*log(x^2)+x^2-3*x)/log((6-2*x)/(log(x^2)+x))^2,x)
Output:
(4*x*( - log(( - 2*x + 6)/(log(x**2) + x)) + 1))/log(( - 2*x + 6)/(log(x** 2) + x))