Optimal. Leaf size=347 \[ \frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}}-\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}} \]
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Rubi [A] time = 0.27, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1414, 1169, 634, 618, 204, 628} \[ \frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2-\sqrt {2}}}-\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2+\sqrt {2}}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1414
Rubi steps
\begin {align*} \int \frac {1-x^4}{1+x^8} \, dx &=\frac {1}{2} \int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx\\ &=\frac {\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {2}}}\\ &=-\left (\frac {1}{8} \sqrt {3-2 \sqrt {2}} \int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx\right )-\frac {1}{8} \sqrt {3-2 \sqrt {2}} \int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {\left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {2}}}+\frac {\left (-1+\sqrt {2}\right ) \int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {2}}}+\frac {\left (-1-\sqrt {2}\right ) \int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {2}}}+\frac {\left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {2}}}+\frac {1}{8} \sqrt {3+2 \sqrt {2}} \int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{8} \sqrt {3+2 \sqrt {2}} \int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx\\ &=\frac {1}{8} \sqrt {1-\frac {1}{\sqrt {2}}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{8} \sqrt {1-\frac {1}{\sqrt {2}}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{8} \sqrt {1+\frac {1}{\sqrt {2}}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{8} \sqrt {1+\frac {1}{\sqrt {2}}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{4} \sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 x\right )+\frac {1}{4} \sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 x\right )-\frac {1}{4} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 x\right )-\frac {1}{4} \sqrt {3+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )\\ &=-\frac {1}{4} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {1-\frac {1}{\sqrt {2}}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{8} \sqrt {1-\frac {1}{\sqrt {2}}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{8} \sqrt {1+\frac {1}{\sqrt {2}}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{8} \sqrt {1+\frac {1}{\sqrt {2}}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 257, normalized size = 0.74 \[ \frac {1}{8} \left (-\left (\sin \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right )\right ) \log \left (x^2-2 x \cos \left (\frac {\pi }{8}\right )+1\right )+\left (\sin \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right )\right ) \log \left (x^2+2 x \cos \left (\frac {\pi }{8}\right )+1\right )+\left (\sin \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right )\right ) \log \left (x^2+2 x \sin \left (\frac {\pi }{8}\right )+1\right )+\left (\cos \left (\frac {\pi }{8}\right )-\sin \left (\frac {\pi }{8}\right )\right ) \log \left (x^2-2 x \sin \left (\frac {\pi }{8}\right )+1\right )+2 \left (\sin \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (\csc \left (\frac {\pi }{8}\right ) \left (x+\cos \left (\frac {\pi }{8}\right )\right )\right )+2 \left (\sin \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right )+2 \left (\sin \left (\frac {\pi }{8}\right )+\cos \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (x \sec \left (\frac {\pi }{8}\right )-\tan \left (\frac {\pi }{8}\right )\right )+2 \left (\cos \left (\frac {\pi }{8}\right )-\sin \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-x \csc \left (\frac {\pi }{8}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 991, normalized size = 2.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 247, normalized size = 0.71 \[ \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 4} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 4} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {-2 \, \sqrt {2} + 4} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{16} \, \sqrt {-2 \, \sqrt {2} + 4} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 29, normalized size = 0.08 \[ \frac {\left (-\RootOf \left (\textit {\_Z}^{8}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+1\right )^{7}} \]
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 \[ -\int \frac {x^{4} - 1}{x^{8} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 312, normalized size = 0.90 \[ -\ln \left ({\left (\frac {\sqrt {-2\,\sqrt {2}-4}}{16}-\frac {\sqrt {4-2\,\sqrt {2}}}{16}\right )}^3\,\left (65536\,x-16384\,\sqrt {-2\,\sqrt {2}-4}+16384\,\sqrt {4-2\,\sqrt {2}}\right )-256\right )\,\left (\frac {\sqrt {-2\,\sqrt {2}-4}}{16}-\frac {\sqrt {4-2\,\sqrt {2}}}{16}\right )-\mathrm {atan}\left (-\frac {x\,1{}\mathrm {i}}{\sqrt {\sqrt {2}-2}}+\frac {x\,1{}\mathrm {i}}{\sqrt {\sqrt {2}+2}}+\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}-2}}+\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}+2}}\right )\,\left (\frac {\sqrt {2}\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\frac {\mathrm {atan}\left (x\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (\frac {1}{2}+1{}\mathrm {i}\right )+\sqrt {2}\,x\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (-\frac {1}{4}-\frac {3}{4}{}\mathrm {i}\right )\right )\,\left (-2+\sqrt {2}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}+\frac {\mathrm {atan}\left (x\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (1-\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,x\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (-\frac {3}{4}+\frac {1}{4}{}\mathrm {i}\right )\right )\,\left (\sqrt {2}\,\left (1+1{}\mathrm {i}\right )-2{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}+\sqrt {2}\,\ln \left (x+{\left (\sqrt {2}+2\right )}^{3/2}\,\left (-1+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (\frac {3}{4}-\frac {1}{4}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}-2}}{16}+\frac {\sqrt {\sqrt {2}+2}}{16}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.75, size = 20, normalized size = 0.06 \[ - \operatorname {RootSum} {\left (1048576 t^{8} + 1, \left (t \mapsto t \log {\left (4096 t^{5} - 4 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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