\(\int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx\) [4369]

Optimal result

Integrand size = 18, antiderivative size = 13 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=-4 x+\frac {1}{4} \log \left (\log ^2(x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 6820, 2339, 29} \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=\frac {1}{2} \log (\log (x))-4 x \]

[In]

Int[(1 - 8*x*Log[x])/(2*x*Log[x]),x]

[Out]

-4*x + Log[Log[x]]/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=-4 x+\frac {1}{2} \log (\log (x)) \]

[In]

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

[Out]

-4*x + Log[Log[x]]/2

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
default	\(-4 x +\frac {\ln \left (\ln \left (x \right )\right )}{2}\)	\(10\)
norman	\(-4 x +\frac {\ln \left (\ln \left (x \right )\right )}{2}\)	\(10\)
risch	\(-4 x +\frac {\ln \left (\ln \left (x \right )\right )}{2}\)	\(10\)
parallelrisch	\(-4 x +\frac {\ln \left (\ln \left (x \right )\right )}{2}\)	\(10\)
parts	\(-4 x +\frac {\ln \left (\ln \left (x \right )\right )}{2}\)	\(10\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=-4 \, x + \frac {1}{2} \, \log \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

-4*x + 1/2*log(log(x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=- 4 x + \frac {\log {\left (\log {\left (x \right )} \right )}}{2} \]

[In]

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

[Out]

-4*x + log(log(x))/2

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=-4 \, x + \frac {1}{2} \, \log \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

-4*x + 1/2*log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=-4 \, x + \frac {1}{2} \, \log \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

-4*x + 1/2*log(log(x))

Mupad [B] (verification not implemented)

Time = 11.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1-8 x \log (x)}{2 x \log (x)} \, dx=\frac {\ln \left (\ln \left (x\right )\right )}{2}-4\,x \]

[In]

int(-(4*x*log(x) - 1/2)/(x*log(x)),x)

[Out]

log(log(x))/2 - 4*x