Integrand size = 93, antiderivative size = 23 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\log \left (-3-\frac {2}{x}+\frac {2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{x}\right ) \]
Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\log (x)+\log \left (2+3 x-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )\right ) \]
Integrate[(4 + 2*Log[x^2] + 2*Log[x^2]*Log[x*Log[x^2]] - 2*Log[x^2]*Log[x* Log[x^2]]*Log[3*Log[x*Log[x^2]]])/((-2*x - 3*x^2)*Log[x^2]*Log[x*Log[x^2]] + 2*x*Log[x^2]*Log[x*Log[x^2]]*Log[3*Log[x*Log[x^2]]]),x]
Time = 1.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7292, 27, 25, 7293, 2009}
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.
\(\displaystyle \int \frac {2 \log \left (x \log \left (x^2\right )\right ) \log \left (x^2\right )-2 \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right ) \log \left (x^2\right )+2 \log \left (x^2\right )+4}{\left (-3 x^2-2 x\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right ) \log \left (x \log \left (x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (-\log \left (x \log \left (x^2\right )\right ) \log \left (x^2\right )+\log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right ) \log \left (x^2\right )-\log \left (x^2\right )-2\right )}{x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \left (-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )+3 x+2\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\log \left (x \log \left (x^2\right )\right ) \log \left (x^2\right )-\log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right ) \log \left (x^2\right )+\log \left (x^2\right )+2}{x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \left (3 x-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )+2\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\log \left (x \log \left (x^2\right )\right ) \log \left (x^2\right )-\log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right ) \log \left (x^2\right )+\log \left (x^2\right )+2}{x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \left (3 x-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )+2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {-3 x \log \left (x \log \left (x^2\right )\right ) \log \left (x^2\right )+2 \log \left (x^2\right )+4}{2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \left (3 x-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )+2\right )}+\frac {1}{2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\frac {\log (x)}{2}-\frac {1}{2} \log \left (-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )+3 x+2\right )\right )\) |
Int[(4 + 2*Log[x^2] + 2*Log[x^2]*Log[x*Log[x^2]] - 2*Log[x^2]*Log[x*Log[x^ 2]]*Log[3*Log[x*Log[x^2]]])/((-2*x - 3*x^2)*Log[x^2]*Log[x*Log[x^2]] + 2*x *Log[x^2]*Log[x*Log[x^2]]*Log[3*Log[x*Log[x^2]]]),x]
3.11.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 2.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\ln \left (x \right )+\ln \left (-\frac {2 \ln \left (3 \ln \left (x \ln \left (x^{2}\right )\right )\right )}{3}+x +\frac {2}{3}\right )\) | \(22\) |
int((-2*ln(x^2)*ln(x*ln(x^2))*ln(3*ln(x*ln(x^2)))+2*ln(x^2)*ln(x*ln(x^2))+ 2*ln(x^2)+4)/(2*x*ln(x^2)*ln(x*ln(x^2))*ln(3*ln(x*ln(x^2)))+(-3*x^2-2*x)*l n(x^2)*ln(x*ln(x^2))),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\frac {1}{2} \, \log \left (x^{2}\right ) + \log \left (-3 \, x + 2 \, \log \left (3 \, \log \left (x \log \left (x^{2}\right )\right )\right ) - 2\right ) \]
integrate((-2*log(x^2)*log(x*log(x^2))*log(3*log(x*log(x^2)))+2*log(x^2)*l og(x*log(x^2))+2*log(x^2)+4)/(2*x*log(x^2)*log(x*log(x^2))*log(3*log(x*log (x^2)))+(-3*x^2-2*x)*log(x^2)*log(x*log(x^2))),x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=- \log {\left (x \right )} + \log {\left (- \frac {3 x}{2} + \log {\left (3 \log {\left (x \log {\left (x^{2} \right )} \right )} \right )} - 1 \right )} \]
integrate((-2*ln(x**2)*ln(x*ln(x**2))*ln(3*ln(x*ln(x**2)))+2*ln(x**2)*ln(x *ln(x**2))+2*ln(x**2)+4)/(2*x*ln(x**2)*ln(x*ln(x**2))*ln(3*ln(x*ln(x**2))) +(-3*x**2-2*x)*ln(x**2)*ln(x*ln(x**2))),x)
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\log \left (x\right ) + \log \left (-\frac {3}{2} \, x + \log \left (3\right ) + \log \left (\log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) - 1\right ) \]
integrate((-2*log(x^2)*log(x*log(x^2))*log(3*log(x*log(x^2)))+2*log(x^2)*l og(x*log(x^2))+2*log(x^2)+4)/(2*x*log(x^2)*log(x*log(x^2))*log(3*log(x*log (x^2)))+(-3*x^2-2*x)*log(x^2)*log(x*log(x^2))),x, algorithm=\
\[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\int { -\frac {2 \, {\left (\log \left (x^{2}\right ) \log \left (x \log \left (x^{2}\right )\right ) \log \left (3 \, \log \left (x \log \left (x^{2}\right )\right )\right ) - \log \left (x^{2}\right ) \log \left (x \log \left (x^{2}\right )\right ) - \log \left (x^{2}\right ) - 2\right )}}{2 \, x \log \left (x^{2}\right ) \log \left (x \log \left (x^{2}\right )\right ) \log \left (3 \, \log \left (x \log \left (x^{2}\right )\right )\right ) - {\left (3 \, x^{2} + 2 \, x\right )} \log \left (x^{2}\right ) \log \left (x \log \left (x^{2}\right )\right )} \,d x } \]
integrate((-2*log(x^2)*log(x*log(x^2))*log(3*log(x*log(x^2)))+2*log(x^2)*l og(x*log(x^2))+2*log(x^2)+4)/(2*x*log(x^2)*log(x*log(x^2))*log(3*log(x*log (x^2)))+(-3*x^2-2*x)*log(x^2)*log(x*log(x^2))),x, algorithm=\
integrate(-2*(log(x^2)*log(x*log(x^2))*log(3*log(x*log(x^2))) - log(x^2)*l og(x*log(x^2)) - log(x^2) - 2)/(2*x*log(x^2)*log(x*log(x^2))*log(3*log(x*l og(x^2))) - (3*x^2 + 2*x)*log(x^2)*log(x*log(x^2))), x)
Time = 12.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\ln \left (\ln \left (3\right )-\frac {3\,x}{2}+\ln \left (\ln \left (x\,\ln \left (x^2\right )\right )\right )-1\right )-\ln \left (x\right ) \]