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 = 0.0099368, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 2448
Rule 321
Rule 207
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*}
Mathematica [A] time = 0.0023323, size = 25, normalized size = 1. \[ x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac{x}{a}\right )-2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 32, normalized size = 1.3 \begin{align*} x\ln \left ( -{a}^{2}+{x}^{2} \right ) -2\,x-a\ln \left ( -a+x \right ) +a\ln \left ( a+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938799, size = 42, normalized size = 1.68 \begin{align*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1013, size = 76, normalized size = 3.04 \begin{align*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.306389, size = 29, normalized size = 1.16 \begin{align*} - 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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07971, size = 45, normalized size = 1.8 \begin{align*} 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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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