Optimal. Leaf size=29 \[ \frac{1}{2} \log \left (a+b x^2+x\right )-\frac{1}{2} \log \left (a+b x^2-x\right ) \]
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Rubi [A] time = 0.0255999, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1164, 628} \[ \frac{1}{2} \log \left (a+b x^2+x\right )-\frac{1}{2} \log \left (a+b x^2-x\right ) \]
Antiderivative was successfully verified.
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Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{a-b x^2}{a^2+(-1+2 a b) x^2+b^2 x^4} \, dx &=-\left (\frac{1}{2} \int \frac{\frac{1}{b}+2 x}{-\frac{a}{b}-\frac{x}{b}-x^2} \, dx\right )-\frac{1}{2} \int \frac{\frac{1}{b}-2 x}{-\frac{a}{b}+\frac{x}{b}-x^2} \, dx\\ &=-\frac{1}{2} \log \left (a-x+b x^2\right )+\frac{1}{2} \log \left (a+x+b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0177935, size = 29, normalized size = 1. \[ \frac{1}{2} \log \left (a+b x^2+x\right )-\frac{1}{2} \log \left (a+b x^2-x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 26, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( b{x}^{2}+a-x \right ) }{2}}+{\frac{\ln \left ( b{x}^{2}+a+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962186, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left (b x^{2} + a + x\right ) - \frac{1}{2} \, \log \left (b x^{2} + a - x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28171, size = 66, normalized size = 2.28 \begin{align*} \frac{1}{2} \, \log \left (b x^{2} + a + x\right ) - \frac{1}{2} \, \log \left (b x^{2} + a - x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.50626, size = 26, normalized size = 0.9 \begin{align*} - \frac{\log{\left (\frac{a}{b} + x^{2} - \frac{x}{b} \right )}}{2} + \frac{\log{\left (\frac{a}{b} + x^{2} + \frac{x}{b} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2268, size = 34, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left (b x^{2} + a + x\right ) - \frac{1}{2} \, \log \left (b x^{2} + a - x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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