Integrand size = 52, antiderivative size = 31 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=3+\log \left (\frac {x}{5+e^{e^5}}\right )-\log \left (\log (x)+\frac {1}{3} \left (2+\log \left (\log \left (x^2\right )\right )\right )\right ) \] Output:
ln(x/(exp(exp(5))+5))+3-ln(1/3*ln(ln(x^2))+2/3+ln(x))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log (x)-\log \left (4+6 \left (\log (x)-\frac {\log \left (x^2\right )}{2}\right )+3 \log \left (x^2\right )+2 \log \left (\log \left (x^2\right )\right )\right ) \] Input:
Integrate[(-2 + (-1 + 3*Log[x])*Log[x^2] + Log[x^2]*Log[Log[x^2]])/((2*x + 3*x*Log[x])*Log[x^2] + x*Log[x^2]*Log[Log[x^2]]),x]
Output:
Log[x] - Log[4 + 6*(Log[x] - Log[x^2]/2) + 3*Log[x^2] + 2*Log[Log[x^2]]]
Time = 0.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {7292, 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.
\(\displaystyle \int \frac {(3 \log (x)-1) \log \left (x^2\right )+\log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )-2}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(3 \log (x)-1) \log \left (x^2\right )+\log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )-2}{x \log \left (x^2\right ) \left (\log \left (\log \left (x^2\right )\right )+3 \log (x)+2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-3 \log \left (x^2\right )-2}{x \log \left (x^2\right ) \left (\log \left (\log \left (x^2\right )\right )+3 \log (x)+2\right )}+\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (x)-\log \left (\log \left (\log \left (x^2\right )\right )+3 \log (x)+2\right )\) |
Input:
Int[(-2 + (-1 + 3*Log[x])*Log[x^2] + Log[x^2]*Log[Log[x^2]])/((2*x + 3*x*L og[x])*Log[x^2] + x*Log[x^2]*Log[Log[x^2]]),x]
Output:
Log[x] - Log[2 + 3*Log[x] + Log[Log[x^2]]]
Time = 2.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\ln \left (x \right )-\ln \left (\frac {\ln \left (\ln \left (x^{2}\right )\right )}{3}+\frac {2}{3}+\ln \left (x \right )\right )\) | \(18\) |
risch | \(\ln \left (x \right )-\ln \left (3 \ln \left (x \right )+\ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+2\right )\) | \(47\) |
Input:
int((ln(x^2)*ln(ln(x^2))+(3*ln(x)-1)*ln(x^2)-2)/(x*ln(x^2)*ln(ln(x^2))+(3* x*ln(x)+2*x)*ln(x^2)),x,method=_RETURNVERBOSE)
Output:
ln(x)-ln(1/3*ln(ln(x^2))+2/3+ln(x))
Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log \left (x\right ) - \log \left (3 \, \log \left (x\right ) + \log \left (2 \, \log \left (x\right )\right ) + 2\right ) \] Input:
integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log (log(x^2))+(3*x*log(x)+2*x)*log(x^2)),x, algorithm="fricas")
Output:
log(x) - log(3*log(x) + log(2*log(x)) + 2)
Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log {\left (x \right )} - \log {\left (3 \log {\left (x \right )} + \log {\left (2 \log {\left (x \right )} \right )} + 2 \right )} \] Input:
integrate((ln(x**2)*ln(ln(x**2))+(3*ln(x)-1)*ln(x**2)-2)/(x*ln(x**2)*ln(ln (x**2))+(3*x*ln(x)+2*x)*ln(x**2)),x)
Output:
log(x) - log(3*log(x) + log(2*log(x)) + 2)
Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\log \left (x\right ) - \log \left (\log \left (2\right ) + 3 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right ) + 2\right ) \] Input:
integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log (log(x^2))+(3*x*log(x)+2*x)*log(x^2)),x, algorithm="maxima")
Output:
log(x) - log(log(2) + 3*log(x) + log(log(x)) + 2)
\[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\int { \frac {{\left (3 \, \log \left (x\right ) - 1\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) - 2}{x \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + {\left (3 \, x \log \left (x\right ) + 2 \, x\right )} \log \left (x^{2}\right )} \,d x } \] Input:
integrate((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log (log(x^2))+(3*x*log(x)+2*x)*log(x^2)),x, algorithm="giac")
Output:
integrate(((3*log(x) - 1)*log(x^2) + log(x^2)*log(log(x^2)) - 2)/(x*log(x^ 2)*log(log(x^2)) + (3*x*log(x) + 2*x)*log(x^2)), x)
Time = 4.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\ln \left (x\right )-\ln \left (\ln \left (\ln \left (x^2\right )\right )+3\,\ln \left (x\right )+2\right ) \] Input:
int((log(x^2)*(3*log(x) - 1) + log(x^2)*log(log(x^2)) - 2)/(log(x^2)*(2*x + 3*x*log(x)) + x*log(x^2)*log(log(x^2))),x)
Output:
log(x) - log(log(log(x^2)) + 3*log(x) + 2)
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-2+(-1+3 \log (x)) \log \left (x^2\right )+\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{(2 x+3 x \log (x)) \log \left (x^2\right )+x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )+3 \,\mathrm {log}\left (x \right )+2\right )+\mathrm {log}\left (x \right ) \] Input:
int((log(x^2)*log(log(x^2))+(3*log(x)-1)*log(x^2)-2)/(x*log(x^2)*log(log(x ^2))+(3*x*log(x)+2*x)*log(x^2)),x)
Output:
- log(log(log(x**2)) + 3*log(x) + 2) + log(x)