3.41.65 $\int \frac{1}{x+\log (\frac{\log (5)}{4})}dx$

Optimal. Leaf size=17 $$\log (11)+\log \left(\frac{2}{5}(x+\log \left(\frac{\log (5)}{4}\right))\right)$$

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.59, number of steps used = 1, number of rules used = 1, integrand size = 11, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.091, Rules used = {31} $$\begin{array}{c}\log (x+\log \left(\frac{\log (5)}{4}\right))\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(x + Log[Log[5]/4])^(-1),x]

[Out]

Log[x + Log[Log[5]/4]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\log (x+\log \left(\frac{\log (5)}{4}\right))\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.59 $$\begin{array}{c}\log (x+\log \left(\frac{\log (5)}{4}\right))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(x + Log[Log[5]/4])^(-1),x]

[Out]

Log[x + Log[Log[5]/4]]

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fricas [A]  time = 0.57, size = 8, normalized size = 0.47 $$\begin{array}{c}\log (x+\log (\frac{1}{4}\log (5)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(log(1/4*log(5))+x),x, algorithm="fricas")

[Out]

log(x + log(1/4*log(5)))

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giac [A]  time = 0.15, size = 9, normalized size = 0.53 $$\begin{array}{c}\log \left(|x+\log (\frac{1}{4}\log (5))|\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(log(1/4*log(5))+x),x, algorithm="giac")

[Out]

log(abs(x + log(1/4*log(5))))

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maple [A]  time = 0.13, size = 9, normalized size = 0.53

method	result	size
default	$\ln (\ln \left(\frac{\ln (5)}{4}\right)+x)$	$9$
norman	$\ln (\ln \left(\frac{\ln (5)}{4}\right)+x)$	$9$
risch	$\ln (-2\ln (2)+\ln (\ln (5))+x)$	$11$
meijerg	$\ln (1+\frac{x}{\ln \left(\frac{\ln (5)}{4}\right)})$	$13$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(ln(1/4*ln(5))+x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 8, normalized size = 0.47 $$\begin{array}{c}\log (x+\log (\frac{1}{4}\log (5)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(log(1/4*log(5))+x),x, algorithm="maxima")

[Out]

log(x + log(1/4*log(5)))

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mupad [B]  time = 0.09, size = 8, normalized size = 0.47 $$\begin{array}{c}\ln (x+\ln \left(\frac{\ln (5)}{4}\right))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + log(log(5)/4)),x)

[Out]

log(x + log(log(5)/4))

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sympy [A]  time = 0.07, size = 12, normalized size = 0.71 $$\begin{array}{c}\log (x-2\log (2)+\log (\log (5)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(ln(1/4*ln(5))+x),x)

[Out]

log(x - 2*log(2) + log(log(5)))

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