Integrand size = 16, antiderivative size = 30 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=-2 x+2 a \text {arctanh}\left (\frac {x}{a}\right )+\frac {1}{2} x \log \left (\left (-a^2+x^2\right )^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 2498, 327, 213} \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=\frac {1}{2} x \log \left (\left (x^2-a^2\right )^2\right )+2 a \text {arctanh}\left (\frac {x}{a}\right )-2 x \]
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Rule 12
Rule 213
Rule 327
Rule 2498
Rubi steps \begin{align*} \text {integral}& = \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 \text {arctanh}\left (\frac {x}{a}\right )+\frac {1}{2} x \log \left (\left (-a^2+x^2\right )^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=\frac {1}{2} \left (-4 x+4 a \text {arctanh}\left (\frac {x}{a}\right )+x \log \left (\left (a^2-x^2\right )^2\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {x \ln \left (\left (a^{2}-x^{2}\right )^{2}\right )}{2}-2 x -a \ln \left (a -x \right )+a \ln \left (a +x \right )\) | \(35\) |
risch | \(\frac {x \ln \left (\left (-a^{2}+x^{2}\right )^{2}\right )}{2}-2 x +a \ln \left (a +x \right )-a \ln \left (-a +x \right )\) | \(35\) |
parts | \(\frac {x \ln \left (\left (-a^{2}+x^{2}\right )^{2}\right )}{2}-2 x -a \ln \left (a -x \right )+a \ln \left (a +x \right )\) | \(35\) |
parallelrisch | \(-2 a \ln \left (-a +x \right )+\frac {x \ln \left (\left (a^{2}-x^{2}\right )^{2}\right )}{2}+\frac {\ln \left (\left (a^{2}-x^{2}\right )^{2}\right ) a}{2}-2 x\) | \(44\) |
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Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=\frac {1}{2} \, x \log \left (a^{4} - 2 \, a^{2} x^{2} + x^{4}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
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Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=- 2 a \left (\frac {\log {\left (- a + x \right )}}{2} - \frac {\log {\left (a + x \right )}}{2}\right ) + \frac {x \log {\left (\left (- a^{2} + x^{2}\right )^{2} \right )}}{2} - 2 x \]
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Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=\frac {1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=\frac {1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{2}\right ) + a \log \left ({\left | a + x \right |}\right ) - a \log \left ({\left | -a + x \right |}\right ) - 2 \, x \]
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Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx=2\,a\,\mathrm {atanh}\left (\frac {x}{a}\right )-2\,x+\frac {x\,\ln \left ({\left (a^2-x^2\right )}^2\right )}{2} \]
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