\(\int \frac {1+\log (-\frac {x}{\log (16)})+\log (-\frac {x}{\log (16)}) \log (x \log (-\frac {x}{\log (16)}))}{x \log (-\frac {x}{\log (16)}) \log (x \log (-\frac {x}{\log (16)}))} \, dx\) [7522]

Optimal result

Integrand size = 57, antiderivative size = 21 \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\log \left (x \log \left (x-x \left (1-\log \left (-\frac {x}{\log (16)}\right )\right )\right )\right ) \]

[Out]

ln(ln(x-x*(1-ln(-1/4*x/ln(2))))*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6817} \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\log \left (x \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )\right ) \]

[In]

Int[(1 + Log[-(x/Log[16])] + Log[-(x/Log[16])]*Log[x*Log[-(x/Log[16])]])/(x*Log[-(x/Log[16])]*Log[x*Log[-(x/Lo
g[16])]]),x]

[Out]

Log[x*Log[x*Log[-(x/Log[16])]]]

Rule 6817

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[q]]

Rubi steps \begin{align*} \text {integral}& = \log \left (x \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\log \left (-\frac {x}{\log (16)}\right )+\log \left (\log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )\right ) \]

[In]

Integrate[(1 + Log[-(x/Log[16])] + Log[-(x/Log[16])]*Log[x*Log[-(x/Log[16])]])/(x*Log[-(x/Log[16])]*Log[x*Log[
-(x/Log[16])]]),x]

[Out]

Log[-(x/Log[16])] + Log[Log[x*Log[-(x/Log[16])]]]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\ln \left (x \right )+\ln \left (\ln \left (x \ln \left (-\frac {x}{4 \ln \left (2\right )}\right )\right )\right )\)	\(16\)
default	\(\ln \left (x \right )+\ln \left (\ln \left (x \left (-2 \ln \left (2\right )-\ln \left (\ln \left (2\right )\right )+\ln \left (-x \right )\right )\right )\right )\)	\(22\)
norman	\(\ln \left (-\frac {x}{4 \ln \left (2\right )}\right )+\ln \left (\ln \left (x \ln \left (-\frac {x}{4 \ln \left (2\right )}\right )\right )\right )\)	\(22\)
parts	\(\ln \left (x \right )+\ln \left (\ln \left (x \left (-2 \ln \left (2\right )-\ln \left (\ln \left (2\right )\right )+\ln \left (-x \right )\right )\right )\right )\)	\(22\)

[In]

int((ln(-1/4*x/ln(2))*ln(x*ln(-1/4*x/ln(2)))+ln(-1/4*x/ln(2))+1)/x/ln(-1/4*x/ln(2))/ln(x*ln(-1/4*x/ln(2))),x,m
ethod=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(x*ln(-1/4*x/ln(2))))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\log \left (-\frac {x}{4 \, \log \left (2\right )}\right ) + \log \left (\log \left (x \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )\right )\right ) \]

[In]

integrate((log(-1/4*x/log(2))*log(x*log(-1/4*x/log(2)))+log(-1/4*x/log(2))+1)/x/log(-1/4*x/log(2))/log(x*log(-
1/4*x/log(2))),x, algorithm="fricas")

[Out]

log(-1/4*x/log(2)) + log(log(x*log(-1/4*x/log(2))))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\log {\left (x \right )} + \log {\left (\log {\left (x \log {\left (- \frac {x}{4 \log {\left (2 \right )}} \right )} \right )} \right )} \]

[In]

integrate((ln(-1/4*x/ln(2))*ln(x*ln(-1/4*x/ln(2)))+ln(-1/4*x/ln(2))+1)/x/ln(-1/4*x/ln(2))/ln(x*ln(-1/4*x/ln(2)
)),x)

[Out]

log(x) + log(log(x*log(-x/(4*log(2)))))

Maxima [F]

\[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\int { \frac {\log \left (x \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )\right ) \log \left (-\frac {x}{4 \, \log \left (2\right )}\right ) + \log \left (-\frac {x}{4 \, \log \left (2\right )}\right ) + 1}{x \log \left (x \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )\right ) \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )} \,d x } \]

[In]

integrate((log(-1/4*x/log(2))*log(x*log(-1/4*x/log(2)))+log(-1/4*x/log(2))+1)/x/log(-1/4*x/log(2))/log(x*log(-
1/4*x/log(2))),x, algorithm="maxima")

[Out]

integrate((2*log(2) - log(x) + log(-log(2)) - 1)/(x*(2*log(2) + log(-log(2)))*log(x) - x*log(x)^2 + (x*(2*log(
2) + log(-log(2))) - x*log(x))*log(-2*log(2) + log(-x) - log(log(2)))), x) + log(x)

Giac [F]

\[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\int { \frac {\log \left (x \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )\right ) \log \left (-\frac {x}{4 \, \log \left (2\right )}\right ) + \log \left (-\frac {x}{4 \, \log \left (2\right )}\right ) + 1}{x \log \left (x \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )\right ) \log \left (-\frac {x}{4 \, \log \left (2\right )}\right )} \,d x } \]

[In]

integrate((log(-1/4*x/log(2))*log(x*log(-1/4*x/log(2)))+log(-1/4*x/log(2))+1)/x/log(-1/4*x/log(2))/log(x*log(-
1/4*x/log(2))),x, algorithm="giac")

[Out]

integrate((log(x*log(-1/4*x/log(2)))*log(-1/4*x/log(2)) + log(-1/4*x/log(2)) + 1)/(x*log(x*log(-1/4*x/log(2)))
*log(-1/4*x/log(2))), x)

Mupad [B] (verification not implemented)

Time = 12.77 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1+\log \left (-\frac {x}{\log (16)}\right )+\log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )}{x \log \left (-\frac {x}{\log (16)}\right ) \log \left (x \log \left (-\frac {x}{\log (16)}\right )\right )} \, dx=\ln \left (\ln \left (x\,\ln \left (-\frac {x}{4}\right )-x\,\ln \left (\ln \left (2\right )\right )\right )\right )+\ln \left (x\right ) \]

[In]

int((log(-x/(4*log(2))) + log(-x/(4*log(2)))*log(x*log(-x/(4*log(2)))) + 1)/(x*log(-x/(4*log(2)))*log(x*log(-x
/(4*log(2))))),x)

[Out]

log(log(x*log(-x/4) - x*log(log(2)))) + log(x)