Optimal. Leaf size=39 \[ -\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (x+1)+\frac{1}{6} \log (2 x+1) \]
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Rubi [A] time = 0.0171199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 616, 31} \[ -\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (x+1)+\frac{1}{6} \log (2 x+1) \]
Antiderivative was successfully verified.
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Rule 1161
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1-5 x^2+4 x^4} \, dx &=-\left (\frac{1}{4} \int \frac{1}{-\frac{1}{2}-\frac{x}{2}+x^2} \, dx\right )-\frac{1}{4} \int \frac{1}{-\frac{1}{2}+\frac{x}{2}+x^2} \, dx\\ &=-\left (\frac{1}{6} \int \frac{1}{-1+x} \, dx\right )-\frac{1}{6} \int \frac{1}{-\frac{1}{2}+x} \, dx+\frac{1}{6} \int \frac{1}{\frac{1}{2}+x} \, dx+\frac{1}{6} \int \frac{1}{1+x} \, dx\\ &=-\frac{1}{6} \log (1-2 x)-\frac{1}{6} \log (1-x)+\frac{1}{6} \log (1+x)+\frac{1}{6} \log (1+2 x)\\ \end{align*}
Mathematica [A] time = 0.0056717, size = 31, normalized size = 0.79 \[ \frac{1}{6} \log \left (2 x^2+3 x+1\right )-\frac{1}{6} \log \left (2 x^2-3 x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 30, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ( 2\,x-1 \right ) }{6}}-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ( 1+2\,x \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960764, size = 39, normalized size = 1. \begin{align*} \frac{1}{6} \, \log \left (2 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x - 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37242, size = 72, normalized size = 1.85 \begin{align*} \frac{1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac{1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102137, size = 29, normalized size = 0.74 \begin{align*} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{1}{2} \right )}}{6} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{1}{2} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13605, size = 45, normalized size = 1.15 \begin{align*} \frac{1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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