3.77.17 \(\int \frac {-1-2 x+2 \log (\log (2))}{-x+\log (\log (2))} \, dx\)

Optimal. Leaf size=18 \[ -\frac {52}{25}+e^5+2 x+\log (x-\log (\log (2))) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {43} \begin {gather*} 2 x+\log (x-\log (\log (2))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*x + Log[x - Log[Log[2]]]

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} 2 (x-\log (\log (2)))+\log (x-\log (\log (2))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(x - Log[Log[2]]) + Log[x - Log[Log[2]]]

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fricas [A]  time = 1.41, size = 12, normalized size = 0.67 \begin {gather*} 2 \, x + \log \left (x - \log \left (\log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(x - log(log(2)))

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giac [A]  time = 0.23, size = 13, normalized size = 0.72 \begin {gather*} 2 \, x + \log \left ({\left | x - \log \left (\log \relax (2)\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(abs(x - log(log(2))))

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maple [A]  time = 0.20, size = 13, normalized size = 0.72

method	result	size
default	\(2 x +\ln \left (x -\ln \left (\ln \relax (2)\right )\right )\)	\(13\)
norman	\(2 x +\ln \left (\ln \left (\ln \relax (2)\right )-x \right )\)	\(13\)
risch	\(2 x +\ln \left (x -\ln \left (\ln \relax (2)\right )\right )\)	\(13\)
meijerg	\(-2 \ln \left (\ln \relax (2)\right ) \ln \left (1-\frac {x}{\ln \left (\ln \relax (2)\right )}\right )-2 \ln \left (\ln \relax (2)\right ) \left (-\frac {x}{\ln \left (\ln \relax (2)\right )}-\ln \left (1-\frac {x}{\ln \left (\ln \relax (2)\right )}\right )\right )+\ln \left (1-\frac {x}{\ln \left (\ln \relax (2)\right )}\right )\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 12, normalized size = 0.67 \begin {gather*} 2 \, x + \log \left (x - \log \left (\log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(x - log(log(2)))

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mupad [B]  time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} 2\,x+\ln \left (x-\ln \left (\ln \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(x - log(log(2)))

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sympy [A]  time = 0.10, size = 10, normalized size = 0.56 \begin {gather*} 2 x + \log {\left (x - \log {\left (\log {\relax (2 )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + log(x - log(log(2)))

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