Integrand size = 69, antiderivative size = 22 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{4}+\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \] Output:
3/4+3/(x-ln(9*x*ln(3/x)))
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )} \] Input:
Integrate[(-3 + (3 - 3*x)*Log[3/x])/(x^3*Log[3/x] - 2*x^2*Log[3/x]*Log[9*x *Log[3/x]] + x*Log[3/x]*Log[9*x*Log[3/x]]^2),x]
Output:
3/(x - Log[9*x*Log[3/x]])
Time = 0.42 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {7239, 7237}
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.
\(\displaystyle \int \frac {(3-3 x) \log \left (\frac {3}{x}\right )-3}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-3 (x-1) \log \left (\frac {3}{x}\right )-3}{x \log \left (\frac {3}{x}\right ) \left (x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {3}{x-\log \left (9 x \log \left (\frac {3}{x}\right )\right )}\) |
Input:
Int[(-3 + (3 - 3*x)*Log[3/x])/(x^3*Log[3/x] - 2*x^2*Log[3/x]*Log[9*x*Log[3 /x]] + x*Log[3/x]*Log[9*x*Log[3/x]]^2),x]
Output:
3/(x - Log[9*x*Log[3/x]])
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {3}{x -\ln \left (9 x \ln \left (\frac {3}{x}\right )\right )}\) | \(19\) |
default | \(-\frac {3}{x \left (\frac {\ln \left (x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}+\frac {2 \ln \left (3\right )}{x}-1\right )}\) | \(31\) |
parts | \(-\frac {6 \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right ) \left (\frac {1}{x}-1\right )}{\left (\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}-\ln \left (3\right )-\ln \left (\frac {1}{x}\right )-\frac {1}{x}\right ) x \left (-\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}+\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}-\frac {i \pi \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{3}}{x}+\frac {4 \ln \left (3\right )}{x}+\frac {2 \ln \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {2 \ln \left (\frac {1}{x}\right )}{x}-2\right )}+\frac {6 i}{x^{2} \left (\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}-\ln \left (3\right )-\ln \left (\frac {1}{x}\right )-\frac {1}{x}\right ) \left (\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (i x \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )\right )^{3}}{x}+\frac {4 i \ln \left (3\right )}{x}+\frac {2 i \ln \left (\ln \left (3\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {2 i \ln \left (\frac {1}{x}\right )}{x}-2 i\right )}\) | \(381\) |
Input:
int(((-3*x+3)*ln(3/x)-3)/(x*ln(3/x)*ln(9*x*ln(3/x))^2-2*x^2*ln(3/x)*ln(9*x *ln(3/x))+x^3*ln(3/x)),x,method=_RETURNVERBOSE)
Output:
3/(x-ln(9*x*ln(3/x)))
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x - \log \left (9 \, x \log \left (\frac {3}{x}\right )\right )} \] Input:
integrate(((-3*x+3)*log(3/x)-3)/(x*log(3/x)*log(9*x*log(3/x))^2-2*x^2*log( 3/x)*log(9*x*log(3/x))+x^3*log(3/x)),x, algorithm="fricas")
Output:
3/(x - log(9*x*log(3/x)))
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=- \frac {3}{- x + \log {\left (9 x \log {\left (\frac {3}{x} \right )} \right )}} \] Input:
integrate(((-3*x+3)*ln(3/x)-3)/(x*ln(3/x)*ln(9*x*ln(3/x))**2-2*x**2*ln(3/x )*ln(9*x*ln(3/x))+x**3*ln(3/x)),x)
Output:
-3/(-x + log(9*x*log(3/x)))
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=-\frac {3}{i \, \pi - x + 2 \, \log \left (3\right ) + \log \left (x\right ) + \log \left (-\log \left (3\right ) + \log \left (x\right )\right )} \] Input:
integrate(((-3*x+3)*log(3/x)-3)/(x*log(3/x)*log(9*x*log(3/x))^2-2*x^2*log( 3/x)*log(9*x*log(3/x))+x^3*log(3/x)),x, algorithm="maxima")
Output:
-3/(I*pi - x + 2*log(3) + log(x) + log(-log(3) + log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 284, normalized size of antiderivative = 12.91 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3 \, {\left (x \log \left (3\right ) \log \left (\frac {3}{x}\right ) - x \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (3\right ) \log \left (\frac {3}{x}\right ) + \log \left (x\right ) \log \left (\frac {3}{x}\right ) + \log \left (\frac {3}{x}\right )\right )}}{x^{2} \log \left (3\right ) \log \left (\frac {3}{x}\right ) - 2 \, x \log \left (3\right )^{2} \log \left (\frac {3}{x}\right ) - x^{2} \log \left (x\right ) \log \left (\frac {3}{x}\right ) + x \log \left (3\right ) \log \left (x\right ) \log \left (\frac {3}{x}\right ) + x \log \left (x\right )^{2} \log \left (\frac {3}{x}\right ) - x \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x \log \left (3\right ) \log \left (\frac {3}{x}\right ) + 2 \, \log \left (3\right )^{2} \log \left (\frac {3}{x}\right ) + x \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (3\right ) \log \left (x\right ) \log \left (\frac {3}{x}\right ) - \log \left (x\right )^{2} \log \left (\frac {3}{x}\right ) + \log \left (3\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - \log \left (x\right ) \log \left (\frac {3}{x}\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + x \log \left (3\right ) - 2 \, \log \left (3\right )^{2} - x \log \left (x\right ) + \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - \log \left (3\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) + \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right )} \] Input:
integrate(((-3*x+3)*log(3/x)-3)/(x*log(3/x)*log(9*x*log(3/x))^2-2*x^2*log( 3/x)*log(9*x*log(3/x))+x^3*log(3/x)),x, algorithm="giac")
Output:
3*(x*log(3)*log(3/x) - x*log(x)*log(3/x) - log(3)*log(3/x) + log(x)*log(3/ x) + log(3/x))/(x^2*log(3)*log(3/x) - 2*x*log(3)^2*log(3/x) - x^2*log(x)*l og(3/x) + x*log(3)*log(x)*log(3/x) + x*log(x)^2*log(3/x) - x*log(3)*log(3/ x)*log(log(3/x)) + x*log(x)*log(3/x)*log(log(3/x)) - x*log(3)*log(3/x) + 2 *log(3)^2*log(3/x) + x*log(x)*log(3/x) - log(3)*log(x)*log(3/x) - log(x)^2 *log(3/x) + log(3)*log(3/x)*log(log(3/x)) - log(x)*log(3/x)*log(log(3/x)) + x*log(3) - 2*log(3)^2 - x*log(x) + log(3)*log(x) + log(x)^2 - log(3)*log (log(3/x)) + log(x)*log(log(3/x)))
Time = 3.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=\frac {3}{x-\ln \left (9\,x\,\ln \left (\frac {3}{x}\right )\right )} \] Input:
int(-(log(3/x)*(3*x - 3) + 3)/(x^3*log(3/x) + x*log(3/x)*log(9*x*log(3/x)) ^2 - 2*x^2*log(3/x)*log(9*x*log(3/x))),x)
Output:
3/(x - log(9*x*log(3/x)))
Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-3+(3-3 x) \log \left (\frac {3}{x}\right )}{x^3 \log \left (\frac {3}{x}\right )-2 x^2 \log \left (\frac {3}{x}\right ) \log \left (9 x \log \left (\frac {3}{x}\right )\right )+x \log \left (\frac {3}{x}\right ) \log ^2\left (9 x \log \left (\frac {3}{x}\right )\right )} \, dx=-\frac {3}{\mathrm {log}\left (9 \,\mathrm {log}\left (\frac {3}{x}\right ) x \right )-x} \] Input:
int(((-3*x+3)*log(3/x)-3)/(x*log(3/x)*log(9*x*log(3/x))^2-2*x^2*log(3/x)*l og(9*x*log(3/x))+x^3*log(3/x)),x)
Output:
( - 3)/(log(9*log(3/x)*x) - x)