3.51.28 \(\int \frac {-1+(-2+2 x) \log (4 x)}{x \log (4 x)} \, dx\) [5028]

Optimal. Leaf size=19 \[ 9+2 x-\log \left (-\frac {2}{3} x^2 \log (4 x)\right ) \]

[Out]

2*x+9-ln(-2/3*x^2*ln(4*x))

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Rubi [A]
time = 0.13, antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6874, 45, 2339, 29} \begin {gather*} 2 x-2 \log (x)-\log (\log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*x - 2*Log[x] - Log[Log[4*x]]

Rule 29

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 6874

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} 2 x-2 \log (x)-\log (\log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + (-2 + 2*x)*Log[4*x])/(x*Log[4*x]),x]

[Out]

2*x - 2*Log[x] - Log[Log[4*x]]

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Maple [A]
time = 0.32, size = 18, normalized size = 0.95

method	result	size
risch	\(2 x -2 \ln \left (x \right )-\ln \left (\ln \left (4 x \right )\right )\)	\(16\)
derivativedivides	\(2 x -2 \ln \left (4 x \right )-\ln \left (\ln \left (4 x \right )\right )\)	\(18\)
default	\(2 x -2 \ln \left (4 x \right )-\ln \left (\ln \left (4 x \right )\right )\)	\(18\)
norman	\(2 x -2 \ln \left (4 x \right )-\ln \left (\ln \left (4 x \right )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 15, normalized size = 0.79 \begin {gather*} 2 \, x - 2 \, \log \left (x\right ) - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - 2*log(x) - log(log(4*x))

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Fricas [A]
time = 0.39, size = 17, normalized size = 0.89 \begin {gather*} 2 \, x - 2 \, \log \left (4 \, x\right ) - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - 2*log(4*x) - log(log(4*x))

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Sympy [A]
time = 0.04, size = 14, normalized size = 0.74 \begin {gather*} 2 x - 2 \log {\left (x \right )} - \log {\left (\log {\left (4 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - 2*log(x) - log(log(4*x))

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Giac [A]
time = 0.41, size = 15, normalized size = 0.79 \begin {gather*} 2 \, x - 2 \, \log \left (x\right ) - \log \left (\log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - 2*log(x) - log(log(4*x))

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Mupad [B]
time = 3.82, size = 15, normalized size = 0.79 \begin {gather*} 2\,x-\ln \left (\ln \left (4\,x\right )\right )-2\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x - log(log(4*x)) - 2*log(x)

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