Optimal. Leaf size=27 \[ \frac {6}{17} \log \left (x^2+1\right )-\frac {7}{34} \log (1-4 x)+\frac {3}{17} \tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1629, 635, 203, 260} \[ \frac {6}{17} \log \left (x^2+1\right )-\frac {7}{34} \log (1-4 x)+\frac {3}{17} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1629
Rubi steps
\begin {align*} \int \frac {-1+2 x^2}{(-1+4 x) \left (1+x^2\right )} \, dx &=\int \left (-\frac {14}{17 (-1+4 x)}+\frac {3 (1+4 x)}{17 \left (1+x^2\right )}\right ) \, dx\\ &=-\frac {7}{34} \log (1-4 x)+\frac {3}{17} \int \frac {1+4 x}{1+x^2} \, dx\\ &=-\frac {7}{34} \log (1-4 x)+\frac {3}{17} \int \frac {1}{1+x^2} \, dx+\frac {12}{17} \int \frac {x}{1+x^2} \, dx\\ &=\frac {3}{17} \tan ^{-1}(x)-\frac {7}{34} \log (1-4 x)+\frac {6}{17} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.41 \[ -\frac {7}{34} \log (4 x-1)+\frac {6}{17} \log \left ((4 x-1)^2+2 (4 x-1)+17\right )+\frac {3}{17} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 21, normalized size = 0.78 \[ \frac {3}{17} \, \arctan \relax (x) + \frac {6}{17} \, \log \left (x^{2} + 1\right ) - \frac {7}{34} \, \log \left (4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 22, normalized size = 0.81 \[ \frac {3}{17} \, \arctan \relax (x) + \frac {6}{17} \, \log \left (x^{2} + 1\right ) - \frac {7}{34} \, \log \left ({\left | 4 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.81 \[ \frac {3 \arctan \relax (x )}{17}-\frac {7 \ln \left (4 x -1\right )}{34}+\frac {6 \ln \left (x^{2}+1\right )}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.31, size = 21, normalized size = 0.78 \[ \frac {3}{17} \, \arctan \relax (x) + \frac {6}{17} \, \log \left (x^{2} + 1\right ) - \frac {7}{34} \, \log \left (4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 25, normalized size = 0.93 \[ -\frac {7\,\ln \left (x-\frac {1}{4}\right )}{34}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {6}{17}-\frac {3}{34}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {6}{17}+\frac {3}{34}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 0.96 \[ - \frac {7 \log {\left (x - \frac {1}{4} \right )}}{34} + \frac {6 \log {\left (x^{2} + 1 \right )}}{17} + \frac {3 \operatorname {atan}{\relax (x )}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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