Optimal. Leaf size=21 \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]
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Rubi [A] time = 0.0094333, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin{align*} \int \frac{1}{A^4-A^2 B^2+\left (-A^2+B^2\right ) x^2} \, dx &=\frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0041203, size = 21, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x}{A}\right )}{A \left (A^2-B^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 44, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( A+x \right ) }{ \left ( 2\,{A}^{2}-2\,{B}^{2} \right ) A}}-{\frac{\ln \left ( A-x \right ) }{ \left ( 2\,{A}^{2}-2\,{B}^{2} \right ) A}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931059, size = 53, normalized size = 2.52 \begin{align*} \frac{\log \left (A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} - \frac{\log \left (-A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81271, size = 62, normalized size = 2.95 \begin{align*} \frac{\log \left (A + x\right ) - \log \left (-A + x\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.271437, size = 70, normalized size = 3.33 \begin{align*} - \frac{\log{\left (- \frac{A^{3}}{\left (A - B\right ) \left (A + B\right )} + \frac{A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} + \frac{\log{\left (\frac{A^{3}}{\left (A - B\right ) \left (A + B\right )} - \frac{A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07611, size = 55, normalized size = 2.62 \begin{align*} \frac{\log \left ({\left | A + x \right |}\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} - \frac{\log \left ({\left | -A + x \right |}\right )}{2 \,{\left (A^{3} - A B^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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