Optimal. Leaf size=30 \[ \frac{1}{2} x \log \left (\left (x^2-a^2\right )^2\right )+2 a \tanh ^{-1}\left (\frac{x}{a}\right )-2 x \]
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Rubi [A] time = 0.0129419, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {12, 2448, 321, 207} \[ \frac{1}{2} x \log \left (\left (x^2-a^2\right )^2\right )+2 a \tanh ^{-1}\left (\frac{x}{a}\right )-2 x \]
Antiderivative was successfully verified.
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Rule 12
Rule 2448
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx &=\frac{1}{2} \int \log \left (\left (-a^2+x^2\right )^2\right ) \, dx\\ &=\frac{1}{2} x \log \left (\left (-a^2+x^2\right )^2\right )-2 \int \frac{x^2}{-a^2+x^2} \, dx\\ &=-2 x+\frac{1}{2} x \log \left (\left (-a^2+x^2\right )^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 )+\frac{1}{2} x \log \left (\left (-a^2+x^2\right )^2\right )\\ \end{align*}
Mathematica [A] time = 0.0041781, size = 31, normalized size = 1.03 \[ \frac{1}{2} \left (x \log \left (\left (a^2-x^2\right )^2\right )+4 a \tanh ^{-1}\left (\frac{x}{a}\right )-4 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 35, normalized size = 1.2 \begin{align*}{\frac{x\ln \left ( \left ( -{a}^{2}+{x}^{2} \right ) ^{2} \right ) }{2}}-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 = 1.18068, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{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 = 1.89554, size = 96, normalized size = 3.2 \begin{align*} \frac{1}{2} \, x \log \left (a^{4} - 2 \, a^{2} x^{2} + x^{4}\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 = 0.320544, size = 32, normalized size = 1.07 \begin{align*} - 2 a \left (\frac{\log{\left (- a + x \right )}}{2} - \frac{\log{\left (a + x \right )}}{2}\right ) + \frac{x \log{\left (\left (- a^{2} + x^{2}\right )^{2} \right )}}{2} - 2 x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09654, size = 49, normalized size = 1.63 \begin{align*} \frac{1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{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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