Optimal. Leaf size=13 \[ -\frac {\log \left (-1+\frac {4 x}{3}\right )}{x} \]
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Rubi [A]
time = 0.13, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1607, 6820,
14, 36, 29, 31, 2442} \begin {gather*} -\frac {\log \left (\frac {4 x}{3}-1\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 31
Rule 36
Rule 1607
Rule 2442
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{x^2 (-3+4 x)} \, dx\\ &=\int \frac {\frac {4 x}{3-4 x}+\log \left (-1+\frac {4 x}{3}\right )}{x^2} \, dx\\ &=\int \left (-\frac {4}{x (-3+4 x)}+\frac {\log \left (-1+\frac {4 x}{3}\right )}{x^2}\right ) \, dx\\ &=-\left (4 \int \frac {1}{x (-3+4 x)} \, dx\right )+\int \frac {\log \left (-1+\frac {4 x}{3}\right )}{x^2} \, dx\\ &=-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x}+\frac {4}{3} \int \frac {1}{x} \, dx+\frac {4}{3} \int \frac {1}{x \left (-1+\frac {4 x}{3}\right )} \, dx-\frac {16}{3} \int \frac {1}{-3+4 x} \, dx\\ &=-\frac {4}{3} \log (3-4 x)+\frac {4 \log (x)}{3}-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x}-\frac {4}{3} \int \frac {1}{x} \, dx+\frac {16}{9} \int \frac {1}{-1+\frac {4 x}{3}} \, dx\\ &=-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\log \left (-1+\frac {4 x}{3}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(24\) vs.
\(2(11)=22\).
time = 1.32, size = 25, normalized size = 1.92
method | result | size |
norman | \(-\frac {\ln \left (\frac {4 x}{3}-1\right )}{x}\) | \(12\) |
risch | \(-\frac {\ln \left (\frac {4 x}{3}-1\right )}{x}\) | \(12\) |
derivativedivides | \(-\frac {4 \ln \left (\frac {4 x}{3}-1\right )}{3}+\frac {\ln \left (\frac {4 x}{3}-1\right ) \left (\frac {4 x}{3}-1\right )}{x}\) | \(25\) |
default | \(-\frac {4 \ln \left (\frac {4 x}{3}-1\right )}{3}+\frac {\ln \left (\frac {4 x}{3}-1\right ) \left (\frac {4 x}{3}-1\right )}{x}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (11) = 22\).
time = 0.52, size = 31, normalized size = 2.38 \begin {gather*} \frac {{\left (4 \, x - 3\right )} \log \left (4 \, x - 3\right ) + 3 \, \log \left (3\right )}{3 \, x} - \frac {4}{3} \, \log \left (4 \, x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 11, normalized size = 0.85 \begin {gather*} -\frac {\log \left (\frac {4}{3} \, x - 1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.77 \begin {gather*} - \frac {\log {\left (\frac {4 x}{3} - 1 \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 11, normalized size = 0.85 \begin {gather*} -\frac {\log \left (\frac {4}{3} \, x - 1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 11, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (\frac {4\,x}{3}-1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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