3.64.48 \(\int \frac {4-5 \log (4 x)}{x+5 x \log (4 x)} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {x-\frac {5}{9} x (2+\log (4 x))}{3 x^2}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {43} \begin {gather*} \log (5 \log (4 x)+1)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 5*Log[4*x])/(x + 5*x*Log[4*x]),x]

[Out]

-Log[x] + Log[1 + 5*Log[4*x]]

Rule 43

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])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {4-5 x}{1+5 x} \, dx,x,\log (4 x)\right )\\ &=\operatorname {Subst}\left (\int \left (-1+\frac {5}{1+5 x}\right ) \, dx,x,\log (4 x)\right )\\ &=-\log (x)+\log (1+5 \log (4 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 16, normalized size = 0.76 \begin {gather*} -\log (4 x)+\log (1+5 \log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 5*Log[4*x])/(x + 5*x*Log[4*x]),x]

[Out]

-Log[4*x] + Log[1 + 5*Log[4*x]]

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fricas [A]  time = 0.57, size = 16, normalized size = 0.76 \begin {gather*} -\log \left (4 \, x\right ) + \log \left (5 \, \log \left (4 \, x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*log(4*x)+4)/(5*x*log(4*x)+x),x, algorithm="fricas")

[Out]

-log(4*x) + log(5*log(4*x) + 1)

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giac [A]  time = 0.15, size = 14, normalized size = 0.67 \begin {gather*} -\log \relax (x) + \log \left (5 \, \log \left (4 \, x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*log(4*x)+4)/(5*x*log(4*x)+x),x, algorithm="giac")

[Out]

-log(x) + log(5*log(4*x) + 1)

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maple [A]  time = 0.08, size = 13, normalized size = 0.62

method	result	size
risch	\(-\ln \relax (x )+\ln \left (\ln \left (4 x \right )+\frac {1}{5}\right )\)	\(13\)
derivativedivides	\(-\ln \left (4 x \right )+\ln \left (5 \ln \left (4 x \right )+1\right )\)	\(17\)
default	\(-\ln \left (4 x \right )+\ln \left (5 \ln \left (4 x \right )+1\right )\)	\(17\)
norman	\(-\ln \left (4 x \right )+\ln \left (5 \ln \left (4 x \right )+1\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*ln(4*x)+4)/(5*x*ln(4*x)+x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(ln(4*x)+1/5)

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maxima [B]  time = 0.45, size = 57, normalized size = 2.71 \begin {gather*} \frac {1}{5} \, {\left (10 \, \log \relax (2) + 5 \, \log \relax (x) + 1\right )} \log \left (2 \, \log \relax (2) + \log \relax (x) + \frac {1}{5}\right ) - \log \left (4 \, x\right ) \log \left (2 \, \log \relax (2) + \log \relax (x) + \frac {1}{5}\right ) - 2 \, \log \relax (2) - \log \relax (x) + \frac {4}{5} \, \log \left (2 \, \log \relax (2) + \log \relax (x) + \frac {1}{5}\right ) - \frac {1}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*log(4*x)+4)/(5*x*log(4*x)+x),x, algorithm="maxima")

[Out]

1/5*(10*log(2) + 5*log(x) + 1)*log(2*log(2) + log(x) + 1/5) - log(4*x)*log(2*log(2) + log(x) + 1/5) - 2*log(2)
- log(x) + 4/5*log(2*log(2) + log(x) + 1/5) - 1/5

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mupad [B]  time = 4.22, size = 14, normalized size = 0.67 \begin {gather*} \ln \left (5\,\ln \left (4\,x\right )+1\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*log(4*x) - 4)/(x + 5*x*log(4*x)),x)

[Out]

log(5*log(4*x) + 1) - log(x)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.57 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\left (4 x \right )} + \frac {1}{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*ln(4*x)+4)/(5*x*ln(4*x)+x),x)

[Out]

-log(x) + log(log(4*x) + 1/5)

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