Optimal. Leaf size=125 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Rubi [A] time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1419, 1093, 207, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1093
Rule 1419
Rubi steps
\begin {align*} \int \frac {1-x^4}{1-6 x^4+x^8} \, dx &=-\left (\frac {1}{2} \int \frac {1}{-1-2 x^2+x^4} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+2 x^2+x^4} \, dx\\ &=-\frac {\int \frac {1}{-1-\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1}{1-\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{-1+\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{1+\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 114, normalized size = 0.91 \begin {gather*} \frac {\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{1-6 x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.40, size = 199, normalized size = 1.59 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \arctan \left (-x \sqrt {\sqrt {2} + 1} + \sqrt {x^{2} + \sqrt {2} - 1} \sqrt {\sqrt {2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \arctan \left (-x \sqrt {\sqrt {2} - 1} + \sqrt {x^{2} + \sqrt {2} + 1} \sqrt {\sqrt {2} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 135, normalized size = 1.08 \begin {gather*} \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {x}{\sqrt {\sqrt {2} + 1}}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {x}{\sqrt {\sqrt {2} - 1}}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} - 2} \log \left ({\left | x + \sqrt {\sqrt {2} + 1} \right |}\right ) - \frac {1}{16} \, \sqrt {2 \, \sqrt {2} - 2} \log \left ({\left | x - \sqrt {\sqrt {2} + 1} \right |}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x + \sqrt {\sqrt {2} - 1} \right |}\right ) - \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x - \sqrt {\sqrt {2} - 1} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 90, normalized size = 0.72 \begin {gather*} \frac {\sqrt {2}\, \arctanh \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {2}\, \arctanh \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {2}\, \arctan \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{4} - 1}{x^{8} - 6 \, x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 245, normalized size = 1.96 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {1-\sqrt {2}}\,4352{}\mathrm {i}}{3072\,\sqrt {2}-4352}-\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {2}}\,3072{}\mathrm {i}}{3072\,\sqrt {2}-4352}\right )\,\sqrt {1-\sqrt {2}}\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}+4352}+\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {2}-1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}+4352}\right )\,\sqrt {-\sqrt {2}-1}\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}-4352}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}-1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}-4352}\right )\,\sqrt {\sqrt {2}-1}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}+1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}+4352}+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}+4352}\right )\,\sqrt {\sqrt {2}+1}\,1{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 51, normalized size = 0.41 \begin {gather*} - \operatorname {RootSum} {\left (16384 t^{4} - 256 t^{2} - 1, \left (t \mapsto t \log {\left (65536 t^{5} - 28 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (16384 t^{4} + 256 t^{2} - 1, \left (t \mapsto t \log {\left (65536 t^{5} - 28 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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