Integrand size = 18, antiderivative size = 239 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )}{\sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}} \]
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Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {212, 2520, 12, 6139, 6057, 2449, 2352, 2497} \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )}{\sqrt {2}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )}{\sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )}{2 \sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )}{2 \sqrt {2}} \]
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Rule 12
Rule 212
Rule 2352
Rule 2449
Rule 2497
Rule 2520
Rule 6057
Rule 6139
Rubi steps \begin{align*} \text {integral}& = \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+2 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2} \left (1-x^2\right )} \, dx \\ & = \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\sqrt {2} \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx \\ & = \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\sqrt {2} \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 (-1+x)}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 (1+x)}\right ) \, dx \\ & = \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{-1+x} \, dx}{\sqrt {2}}-\frac {\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1+x} \, dx}{\sqrt {2}} \\ & = \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-2 \left (\frac {1}{2} \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {2}}}\right )}{1-\frac {x^2}{2}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {\sqrt {2} (-1+x)}{\left (1-\frac {1}{\sqrt {2}}\right ) \left (1+\frac {x}{\sqrt {2}}\right )}\right )}{1-\frac {x^2}{2}} \, dx+\frac {1}{2} \int \frac {\log \left (\frac {\sqrt {2} (1+x)}{\left (1+\frac {1}{\sqrt {2}}\right ) \left (1+\frac {x}{\sqrt {2}}\right )}\right )}{1-\frac {x^2}{2}} \, dx \\ & = \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}-2 \frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {2}}}\right )}{\sqrt {2}} \\ & = \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\text {Li}_2\left (1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.67 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\frac {\log \left (-1+\sqrt {2}\right ) \log (-1+x)-\log \left (1+\sqrt {2}\right ) \log (-1+x)-\log \left (-1+\sqrt {2}\right ) \log (1+x)+\log \left (1+\sqrt {2}\right ) \log (1+x)-\log \left (\sqrt {2}-x\right ) \log \left (1-x^2\right )+\log \left (\sqrt {2}+x\right ) \log \left (1-x^2\right )+\operatorname {PolyLog}\left (2,-\left (\left (-1+\sqrt {2}\right ) (-1+x)\right )\right )-\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) (-1+x)\right )-\operatorname {PolyLog}\left (2,\left (-1+\sqrt {2}\right ) (1+x)\right )+\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) (1+x)\right )\right )}{2 \sqrt {2}} \]
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Time = 1.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (\ln \left (x -\sqrt {2}\right ) \ln \left (-x^{2}+1\right )-\operatorname {dilog}\left (\frac {x +1}{1+\sqrt {2}}\right )-\ln \left (x -\sqrt {2}\right ) \ln \left (\frac {x +1}{1+\sqrt {2}}\right )-\operatorname {dilog}\left (\frac {-1+x}{\sqrt {2}-1}\right )-\ln \left (x -\sqrt {2}\right ) \ln \left (\frac {-1+x}{\sqrt {2}-1}\right )\right ) \sqrt {2}}{4}+\frac {\left (\ln \left (x +\sqrt {2}\right ) \ln \left (-x^{2}+1\right )-\operatorname {dilog}\left (\frac {x +1}{1-\sqrt {2}}\right )-\ln \left (x +\sqrt {2}\right ) \ln \left (\frac {x +1}{1-\sqrt {2}}\right )-\operatorname {dilog}\left (\frac {-1+x}{-1-\sqrt {2}}\right )-\ln \left (x +\sqrt {2}\right ) \ln \left (\frac {-1+x}{-1-\sqrt {2}}\right )\right ) \sqrt {2}}{4}\) | \(194\) |
risch | \(-\frac {\sqrt {2}\, \ln \left (-x^{2}+1\right ) \ln \left (x -\sqrt {2}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right ) \ln \left (\frac {x +1}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right ) \ln \left (\frac {-1+x}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {x +1}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-1+x}{\sqrt {2}-1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (\frac {x +1}{1-\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (\frac {-1+x}{-1-\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (-x^{2}+1\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-1+x}{-1-\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {x +1}{1-\sqrt {2}}\right )}{4}\) | \(214\) |
parts | \(\frac {\operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right ) \ln \left (-x^{2}+1\right ) \sqrt {2}}{2}+\sqrt {2}\, \left (-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right ) \ln \left (x^{2}-1\right )}{2}+\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}\right )\) | \(276\) |
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\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\int { -\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2} \,d x } \]
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\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=- \int \frac {\log {\left (1 - x^{2} \right )}}{x^{2} - 2}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\frac {1}{4} \, \sqrt {2} {\left ({\left (\log \left (2 \, x + 2 \, \sqrt {2}\right ) - \log \left (2 \, x - 2 \, \sqrt {2}\right )\right )} \log \left (-x^{2} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} + 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} + 1} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} - 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} - 1} + 1\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} + 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} + 1}\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} - 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} - 1}\right )\right )} \]
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\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\int { -\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2} \,d x } \]
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Timed out. \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=-\int \frac {\ln \left (1-x^2\right )}{x^2-2} \,d x \]
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