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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {x}{A}\right )}{A \left (A^2-B^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 27, normalized size = 1.29 \[ \frac {\log \left (A + x\right ) - \log \left (-A + x\right )}{2 \, {\left (A^{3} - A B^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 41, normalized size = 1.95 \[ \frac {\log \left ({\left | A + x \right |}\right )}{2 \, {\left (A^{3} - A B^{2}\right )}} - \frac {\log \left ({\left | -A + x \right |}\right )}{2 \, {\left (A^{3} - A B^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 44, normalized size = 2.10 \[ -\frac {\ln \left (A -x \right )}{2 \left (A^{2}-B^{2}\right ) A}+\frac {\ln \left (A +x \right )}{2 \left (A^{2}-B^{2}\right ) A} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 39, normalized size = 1.86 \[ \frac {\log \left (A + x\right )}{2 \, {\left (A^{3} - A B^{2}\right )}} - \frac {\log \left (-A + x\right )}{2 \, {\left (A^{3} - A B^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 21, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (\frac {x}{A}\right )}{A\,B^2-A^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 70, normalized size = 3.33 \[ - \frac {\log {\left (- \frac {A^{3}}{\left (A - B\right ) \left (A + B\right )} + \frac {A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} + \frac {\log {\left (\frac {A^{3}}{\left (A - B\right ) \left (A + B\right )} - \frac {A B^{2}}{\left (A - B\right ) \left (A + B\right )} + x \right )}}{2 A \left (A - B\right ) \left (A + B\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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