Optimal. Leaf size=20 \[ -5+x+3 \left (2+\log \left (2 x^3 \left (1-4 x^2\right )\right )\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1593, 1802, 260} \begin {gather*} 3 \log \left (1-4 x^2\right )+x+9 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 260
Rule 1593
Rule 1802
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9-x+60 x^2+4 x^3}{x \left (-1+4 x^2\right )} \, dx\\ &=\int \left (1+\frac {9}{x}+\frac {24 x}{-1+4 x^2}\right ) \, dx\\ &=x+9 \log (x)+24 \int \frac {x}{-1+4 x^2} \, dx\\ &=x+9 \log (x)+3 \log \left (1-4 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} x+9 \log (x)+3 \log \left (1-4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 16, normalized size = 0.80 \begin {gather*} x + 3 \, \log \left (4 \, x^{2} - 1\right ) + 9 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 25, normalized size = 1.25 \begin {gather*} x + 3 \, \log \left ({\left | 2 \, x + 1 \right |}\right ) + 3 \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + 9 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 17, normalized size = 0.85
method | result | size |
risch | \(x +9 \ln \relax (x )+3 \ln \left (4 x^{2}-1\right )\) | \(17\) |
default | \(x +3 \ln \left (2 x +1\right )+9 \ln \relax (x )+3 \ln \left (2 x -1\right )\) | \(23\) |
norman | \(x +3 \ln \left (2 x +1\right )+9 \ln \relax (x )+3 \ln \left (2 x -1\right )\) | \(23\) |
meijerg | \(3 \ln \left (-4 x^{2}+1\right )+9 \ln \relax (x )+9 \ln \relax (2)+\frac {9 i \pi }{2}-\frac {i \left (4 i x -2 i \arctanh \left (2 x \right )\right )}{4}+\frac {\arctanh \left (2 x \right )}{2}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 22, normalized size = 1.10 \begin {gather*} x + 3 \, \log \left (2 \, x + 1\right ) + 3 \, \log \left (2 \, x - 1\right ) + 9 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 14, normalized size = 0.70 \begin {gather*} x+3\,\ln \left (x^2-\frac {1}{4}\right )+9\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} x + 9 \log {\relax (x )} + 3 \log {\left (4 x^{2} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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