Optimal. Leaf size=31 \[ -\frac {1}{10} \log \left (4 x^2+1\right )+\frac {1}{5} \log (1-4 x)-\frac {1}{10} \tan ^{-1}(2 x) \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2058, 635, 203, 260} \[ -\frac {1}{10} \log \left (4 x^2+1\right )+\frac {1}{5} \log (1-4 x)-\frac {1}{10} \tan ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 2058
Rubi steps
\begin {align*} \int \frac {1}{-1+4 x-4 x^2+16 x^3} \, dx &=\int \left (\frac {4}{5 (-1+4 x)}+\frac {-1-4 x}{5 \left (1+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{5} \log (1-4 x)+\frac {1}{5} \int \frac {-1-4 x}{1+4 x^2} \, dx\\ &=\frac {1}{5} \log (1-4 x)-\frac {1}{5} \int \frac {1}{1+4 x^2} \, dx-\frac {4}{5} \int \frac {x}{1+4 x^2} \, dx\\ &=-\frac {1}{10} \tan ^{-1}(2 x)+\frac {1}{5} \log (1-4 x)-\frac {1}{10} \log \left (1+4 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {1}{10} \log \left (4 x^2+1\right )+\frac {1}{5} \log (1-4 x)-\frac {1}{10} \tan ^{-1}(2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 25, normalized size = 0.81 \[ -\frac {1}{10} \, \arctan \left (2 \, x\right ) - \frac {1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac {1}{5} \, \log \left (4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 26, normalized size = 0.84 \[ -\frac {1}{10} \, \arctan \left (2 \, x\right ) - \frac {1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac {1}{5} \, \log \left ({\left | 4 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 0.84 \[ -\frac {\arctan \left (2 x \right )}{10}+\frac {\ln \left (4 x -1\right )}{5}-\frac {\ln \left (4 x^{2}+1\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 25, normalized size = 0.81 \[ -\frac {1}{10} \, \arctan \left (2 \, x\right ) - \frac {1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac {1}{5} \, \log \left (4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 25, normalized size = 0.81 \[ \frac {\ln \left (x-\frac {1}{4}\right )}{5}+\ln \left (x-\frac {1}{2}{}\mathrm {i}\right )\,\left (-\frac {1}{10}+\frac {1}{20}{}\mathrm {i}\right )+\ln \left (x+\frac {1}{2}{}\mathrm {i}\right )\,\left (-\frac {1}{10}-\frac {1}{20}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 24, normalized size = 0.77 \[ \frac {\log {\left (x - \frac {1}{4} \right )}}{5} - \frac {\log {\left (x^{2} + \frac {1}{4} \right )}}{10} - \frac {\operatorname {atan}{\left (2 x \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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