\(\int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx\) [6791]

Optimal result

Integrand size = 22, antiderivative size = 17 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=\log (x)-\log \left (5 e^{2 x} \log (2 x)\right ) \]

[Out]

ln(x)-ln(5*exp(x)^2*ln(2*x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6820, 2339, 29} \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=-2 x+\log (x)-\log (\log (2 x)) \]

[In]

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

[Out]

-2*x + Log[x] - Log[Log[2*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}& = \int \left (-2+\frac {1}{x}-\frac {1}{x \log (2 x)}\right ) \, dx \\ & = -2 x+\log (x)-\int \frac {1}{x \log (2 x)} \, dx \\ & = -2 x+\log (x)-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (2 x)\right ) \\ & = -2 x+\log (x)-\log (\log (2 x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=-2 x+\log (x)-\log (\log (2 x)) \]

[In]

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

[Out]

-2*x + Log[x] - Log[Log[2*x]]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
risch	\(\ln \left (x \right )-2 x -\ln \left (\ln \left (2 x \right )\right )\)	\(14\)
parts	\(\ln \left (x \right )-2 x -\ln \left (\ln \left (2 x \right )\right )\)	\(14\)
derivativedivides	\(-2 x +\ln \left (2 x \right )-\ln \left (\ln \left (2 x \right )\right )\)	\(16\)
default	\(-2 x +\ln \left (2 x \right )-\ln \left (\ln \left (2 x \right )\right )\)	\(16\)
norman	\(-2 x +\ln \left (2 x \right )-\ln \left (\ln \left (2 x \right )\right )\)	\(16\)
parallelrisch	\(-2 x +\ln \left (2 x \right )-\ln \left (\ln \left (2 x \right )\right )\)	\(16\)

[In]

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

[Out]

ln(x)-2*x-ln(ln(2*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=-2 \, x + \log \left (2 \, x\right ) - \log \left (\log \left (2 \, x\right )\right ) \]

[In]

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

[Out]

-2*x + log(2*x) - log(log(2*x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=- 2 x + \log {\left (x \right )} - \log {\left (\log {\left (2 x \right )} \right )} \]

[In]

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

[Out]

-2*x + log(x) - log(log(2*x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=-2 \, x + \log \left (x\right ) - \log \left (\log \left (2 \, x\right )\right ) \]

[In]

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

[Out]

-2*x + log(x) - log(log(2*x))

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=-2 \, x + \log \left (x\right ) - \log \left (\log \left (2 \, x\right )\right ) \]

[In]

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

[Out]

-2*x + log(x) - log(log(2*x))

Mupad [B] (verification not implemented)

Time = 11.95 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {-1+(1-2 x) \log (2 x)}{x \log (2 x)} \, dx=\ln \left (x\right )-\ln \left (\ln \left (2\,x\right )\right )-2\,x \]

[In]

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

[Out]

log(x) - log(log(2*x)) - 2*x