\(\int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx\) [7861]

Optimal result

Integrand size = 27, antiderivative size = 11 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-8 x+\frac {x}{\log (\log (x))} \]

[Out]

x/ln(ln(x))-8*x

Rubi [F]

\[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx \]

[In]

Int[(-1 + Log[x]*Log[Log[x]] - 8*Log[x]*Log[Log[x]]^2)/(Log[x]*Log[Log[x]]^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-8 x+\frac {x}{\log (\log (x))} \]

[In]

Integrate[(-1 + Log[x]*Log[Log[x]] - 8*Log[x]*Log[Log[x]]^2)/(Log[x]*Log[Log[x]]^2),x]

[Out]

-8*x + x/Log[Log[x]]

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {x}{\ln \left (\ln \left (x \right )\right )}-8 x\)	\(12\)
norman	\(\frac {x -8 x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}\)	\(15\)
parallelrisch	\(-\frac {8 x \ln \left (\ln \left (x \right )\right )-x}{\ln \left (\ln \left (x \right )\right )}\)	\(18\)

[In]

int((-8*ln(x)*ln(ln(x))^2+ln(x)*ln(ln(x))-1)/ln(x)/ln(ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

x/ln(ln(x))-8*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-\frac {8 \, x \log \left (\log \left (x\right )\right ) - x}{\log \left (\log \left (x\right )\right )} \]

[In]

integrate((-8*log(x)*log(log(x))^2+log(x)*log(log(x))-1)/log(x)/log(log(x))^2,x, algorithm="fricas")

[Out]

-(8*x*log(log(x)) - x)/log(log(x))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=- 8 x + \frac {x}{\log {\left (\log {\left (x \right )} \right )}} \]

[In]

integrate((-8*ln(x)*ln(ln(x))**2+ln(x)*ln(ln(x))-1)/ln(x)/ln(ln(x))**2,x)

[Out]

-8*x + x/log(log(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-8 \, x + \frac {x}{\log \left (\log \left (x\right )\right )} \]

[In]

integrate((-8*log(x)*log(log(x))^2+log(x)*log(log(x))-1)/log(x)/log(log(x))^2,x, algorithm="maxima")

[Out]

-8*x + x/log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-8 \, x + \frac {x}{\log \left (\log \left (x\right )\right )} \]

[In]

integrate((-8*log(x)*log(log(x))^2+log(x)*log(log(x))-1)/log(x)/log(log(x))^2,x, algorithm="giac")

[Out]

-8*x + x/log(log(x))

Mupad [B] (verification not implemented)

Time = 13.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\log (x) \log (\log (x))-8 \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x}{\ln \left (\ln \left (x\right )\right )}-8\,x \]

[In]

int(-(8*log(log(x))^2*log(x) - log(log(x))*log(x) + 1)/(log(log(x))^2*log(x)),x)

[Out]

x/log(log(x)) - 8*x