Optimal. Leaf size=25 \[ -2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right ) \]
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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 = {2498, 327, 213}
\begin {gather*} x \log \left (x^2-a^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 327
Rule 2498
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 \begin {gather*} -2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+x \log \left (-a^2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 32, normalized size = 1.28
method | result | size |
default | \(x \ln \left (-a^{2}+x^{2}\right )-2 x +a \ln \left (a +x \right )-a \ln \left (a -x \right )\) | \(32\) |
risch | \(x \ln \left (-a^{2}+x^{2}\right )-2 x +a \ln \left (a +x \right )-a \ln \left (-a +x \right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.95, size = 31, normalized size = 1.24 \begin {gather*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 31, normalized size = 1.24 \begin {gather*} x \log \left (-a^{2} + x^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 1.16 \begin {gather*} - 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 33, normalized size = 1.32 \begin {gather*} 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 {gather*}
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 \begin {gather*} x\,\ln \left (x^2-a^2\right )-2\,x+2\,a\,\mathrm {atanh}\left (\frac {x}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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