Optimal. Leaf size=14 \[ \log \left (2+\frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right )}\right ) \]
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Rubi [A] time = 0.36, antiderivative size = 27, normalized size of antiderivative = 1.93, number of steps used = 5, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6741, 6692, 36, 29, 31} \begin {gather*} \log \left (2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1\right )-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 6692
Rule 6741
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-x}{x \left (x+\log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right ) \left (1+2 \log \left (x+\log \left (\frac {4}{x}\right )\right )\right )} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{x (1+2 x)} \, dx,x,\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1+2 x} \, dx,x,\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )\\ &=-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )+\log \left (1+2 \log \left (x+\log \left (\frac {4}{x}\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 27, normalized size = 1.93 \begin {gather*} -\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )+\log \left (1+2 \log \left (x+\log \left (\frac {4}{x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 27, normalized size = 1.93 \begin {gather*} \log \left (2 \, \log \left (x + \log \left (\frac {4}{x}\right )\right ) + 1\right ) - \log \left (\log \left (x + \log \left (\frac {4}{x}\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x - 1}{2 \, {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )^{2} + {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1-x}{\left (2 x \ln \left (\frac {4}{x}\right )+2 x^{2}\right ) \ln \left (\ln \left (\frac {4}{x}\right )+x \right )^{2}+\left (x \ln \left (\frac {4}{x}\right )+x^{2}\right ) \ln \left (\ln \left (\frac {4}{x}\right )+x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 29, normalized size = 2.07 \begin {gather*} \log \left (\log \left (x + 2 \, \log \relax (2) - \log \relax (x)\right ) + \frac {1}{2}\right ) - \log \left (\log \left (x + 2 \, \log \relax (2) - \log \relax (x)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.30, size = 71, normalized size = 5.07 \begin {gather*} \ln \left (\frac {\left (2\,\ln \left (x+\ln \left (\frac {4}{x}\right )\right )+1\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right )-\ln \left (\frac {\ln \left (x+\ln \left (\frac {4}{x}\right )\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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