\(\int \frac {-\log (\frac {x}{2}) \log (x)+\log (\frac {x}{2}) (1+\log (\frac {x}{e^4}))+\log (x) (1+\log (\frac {x}{e^4})) \log (\frac {-4-4 \log (\frac {x}{e^4})}{\log (x)})}{\log (x) (x+x \log (\frac {x}{e^4})) \log ^2(\frac {-4-4 \log (\frac {x}{e^4})}{\log (x)})} \, dx\) [998]

Optimal result

Integrand size = 88, antiderivative size = 26 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=\frac {\log \left (\frac {x}{2}\right )}{\log \left (\frac {4 \left (-1-\log \left (\frac {x}{e^4}\right )\right )}{\log (x)}\right )} \]

[Out]

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

Rubi [F]

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

[In]

Int[(-(Log[x/2]*Log[x]) + Log[x/2]*(1 + Log[x/E^4]) + Log[x]*(1 + Log[x/E^4])*Log[(-4 - 4*Log[x/E^4])/Log[x]])
/(Log[x]*(x + x*Log[x/E^4])*Log[(-4 - 4*Log[x/E^4])/Log[x]]^2),x]

[Out]

Defer[Subst][Defer[Int][Log[-4 + 12/x]^(-1), x], x, Log[x]] + Defer[Subst][Defer[Int][(-3*x + Log[8])/((-3 + x
)*x*Log[(12 - 4*x)/x]^2), x], x, Log[x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{x \log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx \\ & = \int \frac {-\log (8)-\log ^2(x) \log \left (-4+\frac {12}{\log (x)}\right )+3 \log (x) \left (1+\log \left (-4+\frac {12}{\log (x)}\right )\right )}{x (3-\log (x)) \log (x) \log ^2\left (-\frac {4 (-3+\log (x))}{\log (x)}\right )} \, dx \\ & = \text {Subst}\left (\int \frac {-3 x+\log (8)-3 x \log \left (-4+\frac {12}{x}\right )+x^2 \log \left (-4+\frac {12}{x}\right )}{(-3+x) x \log ^2\left (-\frac {4 (-3+x)}{x}\right )} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {3 x-\log (8)+3 x \log \left (-4+\frac {12}{x}\right )-x^2 \log \left (-4+\frac {12}{x}\right )}{(3-x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {3 x-\log (8)-(-3+x) x \log \left (-4+\frac {12}{x}\right )}{(3-x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (-4+\frac {12}{x}\right )}+\frac {1}{\log \left (-4+\frac {12}{x}\right )}\right ) \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )+\text {Subst}\left (\int \frac {1}{\log \left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{\log \left (-4+\frac {12}{x}\right )} \, dx,x,\log (x)\right )+\text {Subst}\left (\int \frac {-3 x+\log (8)}{(-3+x) x \log ^2\left (\frac {12-4 x}{x}\right )} \, dx,x,\log (x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=\frac {-\log (8)+3 \log (x)}{3 \log \left (-4+\frac {12}{\log (x)}\right )} \]

[In]

Integrate[(-(Log[x/2]*Log[x]) + Log[x/2]*(1 + Log[x/E^4]) + Log[x]*(1 + Log[x/E^4])*Log[(-4 - 4*Log[x/E^4])/Lo
g[x]])/(Log[x]*(x + x*Log[x/E^4])*Log[(-4 - 4*Log[x/E^4])/Log[x]]^2),x]

[Out]

(-Log[8] + 3*Log[x])/(3*Log[-4 + 12/Log[x]])

Maple [A] (verified)

Time = 6.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {x}{2}\right )}{\ln \left (-\frac {4 \left (\ln \left (x \,{\mathrm e}^{-4}\right )+1\right )}{\ln \left (x \right )}\right )}\)	\(24\)
default	\(\frac {\left (-\ln \left (2\right ) \left (-1+\frac {3}{\ln \left (x \right )}\right )+3-\ln \left (2\right )\right ) \ln \left (x \right )}{6 \ln \left (2\right )+3 \ln \left (-1+\frac {3}{\ln \left (x \right )}\right )}\)	\(39\)
risch	\(-\frac {2 \ln \left (2\right )-2 \ln \left (x \right )}{2 \ln \left (2\right )+i \pi -2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (-6 i+2 i \ln \left (x \right )\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (2 \ln \left (x \right )-6\right ) \operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{2}-i \pi \,\operatorname {csgn}\left (2 \ln \left (x \right )-6\right ) \operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )-i \pi \operatorname {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i \left (-6 i+2 i \ln \left (x \right )\right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {-6 i+2 i \ln \left (x \right )}{\ln \left (x \right )}\right )^{3}}\)	\(265\)

[In]

int(((ln(x/exp(4))+1)*ln(x)*ln((-4*ln(x/exp(4))-4)/ln(x))-ln(1/2*x)*ln(x)+(ln(x/exp(4))+1)*ln(1/2*x))/(x*ln(x/
exp(4))+x)/ln(x)/ln((-4*ln(x/exp(4))-4)/ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=-\frac {\log \left (2\right ) - \log \left (x e^{\left (-4\right )}\right ) - 4}{\log \left (-\frac {4 \, {\left (\log \left (x e^{\left (-4\right )}\right ) + 1\right )}}{\log \left (x e^{\left (-4\right )}\right ) + 4}\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=\frac {\log {\left (x \right )} - \log {\left (2 \right )}}{\log {\left (\frac {12 - 4 \log {\left (x \right )}}{\log {\left (x \right )}} \right )}} \]

[In]

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

[Out]

(log(x) - log(2))/log((12 - 4*log(x))/log(x))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=-\frac {\log \left (2\right ) - \log \left (x\right )}{i \, \pi + 2 \, \log \left (2\right ) + \log \left (\log \left (x\right ) - 3\right ) - \log \left (\log \left (x\right )\right )} \]

[In]

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

[Out]

-(log(2) - log(x))/(I*pi + 2*log(2) + log(log(x) - 3) - log(log(x)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (21) = 42\).

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=-\frac {\frac {{\left (\log \left (x\right ) - 3\right )} \log \left (2\right )}{\log \left (x\right )} - \log \left (2\right ) + 3}{\frac {{\left (\log \left (x\right ) - 3\right )} \log \left (-\frac {4 \, {\left (\log \left (x\right ) - 3\right )}}{\log \left (x\right )}\right )}{\log \left (x\right )} - \log \left (-\frac {4 \, {\left (\log \left (x\right ) - 3\right )}}{\log \left (x\right )}\right )} \]

[In]

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

[Out]

-((log(x) - 3)*log(2)/log(x) - log(2) + 3)/((log(x) - 3)*log(-4*(log(x) - 3)/log(x))/log(x) - log(-4*(log(x) -
3)/log(x)))

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-\log \left (\frac {x}{2}\right ) \log (x)+\log \left (\frac {x}{2}\right ) \left (1+\log \left (\frac {x}{e^4}\right )\right )+\log (x) \left (1+\log \left (\frac {x}{e^4}\right )\right ) \log \left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )}{\log (x) \left (x+x \log \left (\frac {x}{e^4}\right )\right ) \log ^2\left (\frac {-4-4 \log \left (\frac {x}{e^4}\right )}{\log (x)}\right )} \, dx=\frac {\ln \left (\frac {x}{2}\right )}{\ln \left (-\frac {4\,\ln \left (x\right )-12}{\ln \left (x\right )}\right )} \]

[In]

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

[Out]

log(x/2)/log(-(4*log(x) - 12)/log(x))