Integrand size = 109, antiderivative size = 27 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log \left (3 \left (-4-\log \left (\log \left (\frac {2 x^2}{3-x+\frac {\log (x)}{x}}\right )\right )\right )\right ) \] Output:
ln(-12-3*ln(ln(2*x^2/(3+ln(x)/x-x))))
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log \left (4+\log \left (\log \left (\frac {2 x^3}{-((-3+x) x)+\log (x)}\right )\right )\right ) \] Input:
Integrate[(-1 + 6*x - x^2 + 3*Log[x])/((12*x^2 - 4*x^3 + 4*x*Log[x])*Log[( 2*x^3)/(3*x - x^2 + Log[x])] + (3*x^2 - x^3 + x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])]*Log[Log[(2*x^3)/(3*x - x^2 + Log[x])]]),x]
Output:
Log[4 + Log[Log[(2*x^3)/(-((-3 + x)*x) + Log[x])]]]
Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {7292, 7235}
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.
\(\displaystyle \int \frac {-x^2+6 x+3 \log (x)-1}{\left (-4 x^3+12 x^2+4 x \log (x)\right ) \log \left (\frac {2 x^3}{-x^2+3 x+\log (x)}\right )+\left (-x^3+3 x^2+x \log (x)\right ) \log \left (\log \left (\frac {2 x^3}{-x^2+3 x+\log (x)}\right )\right ) \log \left (\frac {2 x^3}{-x^2+3 x+\log (x)}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x^2+6 x+3 \log (x)-1}{x \left (-x^2+3 x+\log (x)\right ) \log \left (\frac {2 x^3}{\log (x)-(x-3) x}\right ) \left (\log \left (\log \left (\frac {2 x^3}{\log (x)-(x-3) x}\right )\right )+4\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (\log \left (\log \left (\frac {2 x^3}{(3-x) x+\log (x)}\right )\right )+4\right )\) |
Input:
Int[(-1 + 6*x - x^2 + 3*Log[x])/((12*x^2 - 4*x^3 + 4*x*Log[x])*Log[(2*x^3) /(3*x - x^2 + Log[x])] + (3*x^2 - x^3 + x*Log[x])*Log[(2*x^3)/(3*x - x^2 + Log[x])]*Log[Log[(2*x^3)/(3*x - x^2 + Log[x])]]),x]
Output:
Log[4 + Log[Log[(2*x^3)/((3 - x)*x + Log[x])]]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 9.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\ln \left (\ln \left (\ln \left (\frac {2 x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )\right )+4\right )\) | \(24\) |
default | \(\ln \left (\ln \left (\ln \left (2\right )+3 \ln \left (x \right )-\ln \left (\ln \left (x \right )-x^{2}+3 x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right ) \left (-\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\operatorname {csgn}\left (i x^{3}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x^{2}+3 x}\right )\right )}{2}\right )+4\right )\) | \(191\) |
risch | \(\ln \left (\ln \left (\ln \left (2\right )+i \pi +3 \ln \left (x \right )-\ln \left (-\ln \left (x \right )+x^{2}-3 x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right ) \left (\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )+\operatorname {csgn}\left (i x^{3}\right )\right ) \left (\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )-\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x^{2}+3 x}\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )^{2} \left (-\operatorname {csgn}\left (\frac {i x^{3}}{\ln \left (x \right )-x^{2}+3 x}\right )-1\right )\right )+4\right )\) | \(243\) |
Input:
int((3*ln(x)-x^2+6*x-1)/((x*ln(x)-x^3+3*x^2)*ln(2*x^3/(ln(x)-x^2+3*x))*ln( ln(2*x^3/(ln(x)-x^2+3*x)))+(4*x*ln(x)-4*x^3+12*x^2)*ln(2*x^3/(ln(x)-x^2+3* x))),x,method=_RETURNVERBOSE)
Output:
ln(ln(ln(2*x^3/(ln(x)-x^2+3*x)))+4)
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (-\frac {2 \, x^{3}}{x^{2} - 3 \, x - \log \left (x\right )}\right )\right ) + 4\right ) \] Input:
integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2 +3*x))*log(log(2*x^3/(log(x)-x^2+3*x)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^ 3/(log(x)-x^2+3*x))),x, algorithm="fricas")
Output:
log(log(log(-2*x^3/(x^2 - 3*x - log(x)))) + 4)
Time = 1.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log {\left (\log {\left (\log {\left (\frac {2 x^{3}}{- x^{2} + 3 x + \log {\left (x \right )}} \right )} \right )} + 4 \right )} \] Input:
integrate((3*ln(x)-x**2+6*x-1)/((x*ln(x)-x**3+3*x**2)*ln(2*x**3/(ln(x)-x** 2+3*x))*ln(ln(2*x**3/(ln(x)-x**2+3*x)))+(4*x*ln(x)-4*x**3+12*x**2)*ln(2*x* *3/(ln(x)-x**2+3*x))),x)
Output:
log(log(log(2*x**3/(-x**2 + 3*x + log(x)))) + 4)
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log \left (\log \left (\log \left (2\right ) - \log \left (-x^{2} + 3 \, x + \log \left (x\right )\right ) + 3 \, \log \left (x\right )\right ) + 4\right ) \] Input:
integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2 +3*x))*log(log(2*x^3/(log(x)-x^2+3*x)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^ 3/(log(x)-x^2+3*x))),x, algorithm="maxima")
Output:
log(log(log(2) - log(-x^2 + 3*x + log(x)) + 3*log(x)) + 4)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\log \left (\log \left (i \, \pi + \log \left (2\right ) - \log \left (x^{2} - 3 \, x - \log \left (x\right )\right ) + 3 \, \log \left (x\right )\right ) + 4\right ) \] Input:
integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2 +3*x))*log(log(2*x^3/(log(x)-x^2+3*x)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^ 3/(log(x)-x^2+3*x))),x, algorithm="giac")
Output:
log(log(I*pi + log(2) - log(x^2 - 3*x - log(x)) + 3*log(x)) + 4)
Time = 3.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (\frac {2\,x^3}{3\,x+\ln \left (x\right )-x^2}\right )\right )+4\right ) \] Input:
int((6*x + 3*log(x) - x^2 - 1)/(log((2*x^3)/(3*x + log(x) - x^2))*(4*x*log (x) + 12*x^2 - 4*x^3) + log((2*x^3)/(3*x + log(x) - x^2))*log(log((2*x^3)/ (3*x + log(x) - x^2)))*(x*log(x) + 3*x^2 - x^3)),x)
Output:
log(log(log((2*x^3)/(3*x + log(x) - x^2))) + 4)
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {2 x^{3}}{\mathrm {log}\left (x \right )-x^{2}+3 x}\right )\right )+4\right ) \] Input:
int((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2+3*x)) *log(log(2*x^3/(log(x)-x^2+3*x)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^3/(log (x)-x^2+3*x))),x)
Output:
log(log(log((2*x**3)/(log(x) - x**2 + 3*x))) + 4)