Optimal. Leaf size=25 \[ x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2448, 321, 207} \[ x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2448
Rubi steps
\begin {align*} \int \log \left (-a^2+x^2\right ) \, dx &=x \log \left (-a^2+x^2\right )-2 \int \frac {x^2}{-a^2+x^2} \, dx\\ &=-2 x+x \log \left (-a^2+x^2\right )-\left (2 a^2\right ) \int \frac {1}{-a^2+x^2} \, dx\\ &=-2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 31, normalized size = 1.24 \[ x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 33, normalized size = 1.32 \[ x \log \left (-a^{2} + x^{2}\right ) + a \log \left ({\left | a + x \right |}\right ) - a \log \left ({\left | -a + x \right |}\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 1.28 \[ -a \ln \left (-a +x \right )+a \ln \left (a +x \right )+x \ln \left (-a^{2}+x^{2}\right )-2 x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 31, normalized size = 1.24 \[ x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 25, normalized size = 1.00 \[ x\,\ln \left (x^2-a^2\right )-2\,x+2\,a\,\mathrm {atanh}\left (\frac {x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 29, normalized size = 1.16 \[ - 2 a \left (\frac {\log {\left (- a + x \right )}}{2} - \frac {\log {\left (a + x \right )}}{2}\right ) + x \log {\left (- a^{2} + x^{2} \right )} - 2 x \]
Verification of antiderivative is not currently implemented for this CAS.
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