Optimal. Leaf size=2 \[ \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0020564, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {28, 21, 207} \[ \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 21
Rule 207
Rubi steps
\begin{align*} \int \frac{1-x^2}{1-2 x^2+x^4} \, dx &=\int \frac{1-x^2}{\left (-1+x^2\right )^2} \, dx\\ &=-\int \frac{1}{-1+x^2} \, dx\\ &=\tanh ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0021777, size = 19, normalized size = 9.5 \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (1-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 3, normalized size = 1.5 \begin{align*}{\it Artanh} \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.969496, size = 18, normalized size = 9. \begin{align*} \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32515, size = 45, normalized size = 22.5 \begin{align*} \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.087152, size = 12, normalized size = 6. \begin{align*} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14393, size = 20, normalized size = 10. \begin{align*} \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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