3.81.23 \(\int (3-\log (x)-\log (\frac {\log (2)}{5 x})) \, dx\)

Optimal. Leaf size=22 \[ 5+x \left (3-\log (x)-\log \left (\frac {\log (2)}{5 x}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2295} \begin {gather*} 3 x+x (-\log (x))-x \log \left (\frac {\log (2)}{5 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3 - Log[x] - Log[Log[2]/(5*x)],x]

[Out]

3*x - x*Log[x] - x*Log[Log[2]/(5*x)]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3 x-\int \log (x) \, dx-\int \log \left (\frac {\log (2)}{5 x}\right ) \, dx\\ &=3 x-x \log (x)-x \log \left (\frac {\log (2)}{5 x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} 3 x-x \log (x)-x \log \left (\frac {\log (2)}{5 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3 - Log[x] - Log[Log[2]/(5*x)],x]

[Out]

3*x - x*Log[x] - x*Log[Log[2]/(5*x)]

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fricas [A]  time = 0.77, size = 12, normalized size = 0.55 \begin {gather*} -x \log \left (\frac {1}{5} \, \log \relax (2)\right ) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(1/5*log(2)/x)+3-log(x),x, algorithm="fricas")

[Out]

-x*log(1/5*log(2)) + 3*x

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giac [A]  time = 0.22, size = 15, normalized size = 0.68 \begin {gather*} -x {\left (\log \relax (x) + \log \left (\frac {\log \relax (2)}{5 \, x}\right ) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(1/5*log(2)/x)+3-log(x),x, algorithm="giac")

[Out]

-x*(log(x) + log(1/5*log(2)/x) - 3)

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maple [A]  time = 0.02, size = 21, normalized size = 0.95

method	result	size
default	\(-x \ln \relax (x )+3 x -\ln \left (\frac {\ln \relax (2)}{5 x}\right ) x\)	\(21\)
norman	\(-x \ln \relax (x )+3 x -\ln \left (\frac {\ln \relax (2)}{5 x}\right ) x\)	\(21\)
risch	\(-x \ln \relax (x )+3 x -\ln \left (\frac {\ln \relax (2)}{5 x}\right ) x\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-ln(1/5*ln(2)/x)+3-ln(x),x,method=_RETURNVERBOSE)

[Out]

-x*ln(x)+3*x-ln(1/5*ln(2)/x)*x

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maxima [A]  time = 0.35, size = 20, normalized size = 0.91 \begin {gather*} -x \log \relax (x) - x \log \left (\frac {\log \relax (2)}{5 \, x}\right ) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(1/5*log(2)/x)+3-log(x),x, algorithm="maxima")

[Out]

-x*log(x) - x*log(1/5*log(2)/x) + 3*x

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mupad [B]  time = 4.83, size = 15, normalized size = 0.68 \begin {gather*} -x\,\left (\ln \left (\frac {\ln \relax (2)}{5\,x}\right )+\ln \relax (x)-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3 - log(x) - log(log(2)/(5*x)),x)

[Out]

-x*(log(log(2)/(5*x)) + log(x) - 3)

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sympy [A]  time = 0.20, size = 10, normalized size = 0.45 \begin {gather*} x \left (- \log {\left (\log {\relax (2 )} \right )} + \log {\relax (5 )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-ln(1/5*ln(2)/x)+3-ln(x),x)

[Out]

x*(-log(log(2)) + log(5) + 3)

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