Optimal. Leaf size=27 \[ \frac {1}{4} \log (5) \left (1+\log \left (\frac {1}{2} \left (1+\log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )\right )\right )\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 21, normalized size of antiderivative = 0.78, number of steps
used = 4, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6820, 6816}
\begin {gather*} \frac {1}{4} \log (5) \log \left (\log \left (x-\log \left (\frac {3 x}{4}\right )+1\right )+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6816
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log (5) \int \frac {1-x}{-4 x-4 x^2+4 x \log \left (\frac {3 x}{4}\right )+\left (-4 x-4 x^2+4 x \log \left (\frac {3 x}{4}\right )\right ) \log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )} \, dx\\ &=\log (5) \int \frac {-1+x}{4 x \left (1+x-\log \left (\frac {3 x}{4}\right )\right ) \left (1+\log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )\right )} \, dx\\ &=\frac {1}{4} \log (5) \int \frac {-1+x}{x \left (1+x-\log \left (\frac {3 x}{4}\right )\right ) \left (1+\log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )\right )} \, dx\\ &=\frac {1}{4} \log (5) \log \left (1+\log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{4} \log (5) \log \left (1+\log \left (1+x-\log \left (\frac {3 x}{4}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 13.67, size = 18, normalized size = 0.67
method | result | size |
norman | \(\frac {\ln \left (5\right ) \ln \left (\ln \left (-\ln \left (\frac {3 x}{4}\right )+x +1\right )+1\right )}{4}\) | \(18\) |
risch | \(\frac {\ln \left (5\right ) \ln \left (\ln \left (-\ln \left (\frac {3 x}{4}\right )+x +1\right )+1\right )}{4}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, \log \left (5\right ) \log \left (\log \left (x - \log \left (3\right ) + 2 \, \log \left (2\right ) - \log \left (x\right ) + 1\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{4} \, \log \left (5\right ) \log \left (\log \left (x - \log \left (\frac {3}{4} \, x\right ) + 1\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 19, normalized size = 0.70 \begin {gather*} \frac {\log {\left (5 \right )} \log {\left (\log {\left (x - \log {\left (\frac {3 x}{4} \right )} + 1 \right )} + 1 \right )}}{4} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.56, size = 17, normalized size = 0.63 \begin {gather*} \frac {\ln \left (5\right )\,\ln \left (\ln \left (x-\ln \left (\frac {3\,x}{4}\right )+1\right )+1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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