Optimal. Leaf size=102 \[ -\frac {1}{x}-\frac {1}{4} \tan ^{-1}(x)+\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}(x)-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {331, 307,
303, 1176, 631, 210, 1179, 642, 304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}(x)}{4}+\frac {\text {ArcTan}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\text {ArcTan}\left (\sqrt {2} x+1\right )}{4 \sqrt {2}}-\frac {\log \left (x^2-\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2} x+1\right )}{8 \sqrt {2}}-\frac {1}{x}+\frac {1}{4} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 212
Rule 303
Rule 304
Rule 307
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1-x^8\right )} \, dx &=-\frac {1}{x}+\int \frac {x^6}{1-x^8} \, dx\\ &=-\frac {1}{x}+\frac {1}{2} \int \frac {x^2}{1-x^4} \, dx-\frac {1}{2} \int \frac {x^2}{1+x^4} \, dx\\ &=-\frac {1}{x}+\frac {1}{4} \int \frac {1}{1-x^2} \, dx-\frac {1}{4} \int \frac {1}{1+x^2} \, dx+\frac {1}{4} \int \frac {1-x^2}{1+x^4} \, dx-\frac {1}{4} \int \frac {1+x^2}{1+x^4} \, dx\\ &=-\frac {1}{x}-\frac {1}{4} \tan ^{-1}(x)+\frac {1}{4} \tanh ^{-1}(x)-\frac {1}{8} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx-\frac {\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{8 \sqrt {2}}\\ &=-\frac {1}{x}-\frac {1}{4} \tan ^{-1}(x)+\frac {1}{4} \tanh ^{-1}(x)-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{4 \sqrt {2}}\\ &=-\frac {1}{x}-\frac {1}{4} \tan ^{-1}(x)+\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}(x)-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 109, normalized size = 1.07 \begin {gather*} -\frac {16+4 x \tan ^{-1}(x)-2 \sqrt {2} x \tan ^{-1}\left (1-\sqrt {2} x\right )+2 \sqrt {2} x \tan ^{-1}\left (1+\sqrt {2} x\right )+2 x \log (1-x)-2 x \log (1+x)+\sqrt {2} x \log \left (1-\sqrt {2} x+x^2\right )-\sqrt {2} x \log \left (1+\sqrt {2} x+x^2\right )}{16 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 74, normalized size = 0.73
method | result | size |
risch | \(-\frac {1}{x}+\frac {\ln \left (x +1\right )}{8}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}+x \right )\right )}{8}-\frac {\ln \left (x -1\right )}{8}-\frac {\arctan \left (x \right )}{4}\) | \(44\) |
default | \(\frac {\ln \left (x +1\right )}{8}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x^{2}-x \sqrt {2}}{1+x^{2}+x \sqrt {2}}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{16}-\frac {\ln \left (x -1\right )}{8}-\frac {\arctan \left (x \right )}{4}-\frac {1}{x}\) | \(74\) |
meijerg | \(\frac {\left (-1\right )^{\frac {1}{8}} \left (\frac {8 \left (-1\right )^{\frac {7}{8}}}{x}+\frac {x^{7} \left (-1\right )^{\frac {7}{8}} \left (\ln \left (1-\left (x^{8}\right )^{\frac {1}{8}}\right )-\ln \left (1+\left (x^{8}\right )^{\frac {1}{8}}\right )+\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{2}+\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}{2-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}\right )+2 \arctan \left (\left (x^{8}\right )^{\frac {1}{8}}\right )-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{2}+\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}{2+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{8}}}\right )\right )}{\left (x^{8}\right )^{\frac {7}{8}}}\right )}{8}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 93, normalized size = 0.91 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) - \frac {1}{x} - \frac {1}{4} \, \arctan \left (x\right ) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 128, normalized size = 1.25 \begin {gather*} \frac {4 \, \sqrt {2} x \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + 1} - 1\right ) + 4 \, \sqrt {2} x \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - \sqrt {2} x + 1} + 1\right ) + \sqrt {2} x \log \left (4 \, x^{2} + 4 \, \sqrt {2} x + 4\right ) - \sqrt {2} x \log \left (4 \, x^{2} - 4 \, \sqrt {2} x + 4\right ) - 4 \, x \arctan \left (x\right ) + 2 \, x \log \left (x + 1\right ) - 2 \, x \log \left (x - 1\right ) - 16}{16 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 82.46, size = 49, normalized size = 0.48 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{8} + \frac {\log {\left (x + 1 \right )}}{8} + \frac {i \log {\left (x - i \right )}}{8} - \frac {i \log {\left (x + i \right )}}{8} - \operatorname {RootSum} {\left (4096 t^{4} + 1, \left ( t \mapsto t \log {\left (- 2097152 t^{7} + x \right )} \right )\right )} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 95, normalized size = 0.93 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) - \frac {1}{x} - \frac {1}{4} \, \arctan \left (x\right ) + \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.04, size = 50, normalized size = 0.49 \begin {gather*} -\frac {\mathrm {atan}\left (x\right )}{4}-\frac {1}{x}-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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