Optimal. Leaf size=347 \[ -\frac {1}{4} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}} \]
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Rubi [A]
time = 0.17, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1427, 1108,
648, 632, 210, 642} \begin {gather*} -\frac {1}{4} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {ArcTan}\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 )} \text {ArcTan}\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 )} \text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {\log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2-\sqrt {2}}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2-\sqrt {2}}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2+\sqrt {2}}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2+\sqrt {2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1108
Rule 1427
Rubi steps
\begin {align*} \int \frac {1+x^4}{1+x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {2} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {2} x^2+x^4} \, dx\\ &=\frac {\int \frac {\sqrt {2-\sqrt {2}}-x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2+\sqrt {2}}-x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\sqrt {2+\sqrt {2}}+x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {2}}}\\ &=\frac {1}{8} \int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx-\frac {\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {2}}}\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 x\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 {\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 {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2-\sqrt {2}}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2+\sqrt {2}}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 258, normalized size = 0.74 \begin {gather*} \frac {1}{8} \left (2 \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )-\sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (x \sec \left (\frac {\pi }{8}\right )-\tan \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )-\sin \left (\frac {\pi }{8}\right )\right )+\log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )-\sin \left (\frac {\pi }{8}\right )\right )+\log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right ) \left (-\cos \left (\frac {\pi }{8}\right )+\sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )+\sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )+\sin \left (\frac {\pi }{8}\right )\right )-\log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )+\sin \left (\frac {\pi }{8}\right )\right )+\log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right ) \left (\cos \left (\frac {\pi }{8}\right )+\sin \left (\frac {\pi }{8}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 27, normalized size = 0.08
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8}\) | \(27\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8}\) | \(27\) |
meijerg | \(\text {Expression too large to display}\) | \(566\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1001 vs.
\(2 (245) = 490\).
time = 0.39, size = 1001, normalized size = 2.88 \begin {gather*} -\frac {1}{8} \, {\left (\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} + x \sqrt {-\sqrt {2} + 2} + 1} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} - x \sqrt {-\sqrt {2} + 2} + 1} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} + x \sqrt {\sqrt {2} + 2} + 1} + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, x - 2 \, \sqrt {x^{2} - x \sqrt {\sqrt {2} + 2} + 1} - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {2 \, \sqrt {2} x - 2 \, \sqrt {2 \, x^{2} + \sqrt {2} x \sqrt {\sqrt {2} + 2} - \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 2} + \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {2 \, \sqrt {2} x - 2 \, \sqrt {2 \, x^{2} - \sqrt {2} x \sqrt {\sqrt {2} + 2} + \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 2} - \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {2} x - 2 \, \sqrt {2 \, x^{2} + \sqrt {2} x \sqrt {\sqrt {2} + 2} + \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 2} + \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, \sqrt {2} x - 2 \, \sqrt {2 \, x^{2} - \sqrt {2} x \sqrt {\sqrt {2} + 2} - \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 2} - \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{32} \, {\left (\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}\right )} \log \left (1024 \, x^{2} + 1024 \, x \sqrt {\sqrt {2} + 2} + 1024\right ) - \frac {1}{32} \, {\left (\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}\right )} \log \left (1024 \, x^{2} - 1024 \, x \sqrt {\sqrt {2} + 2} + 1024\right ) + \frac {1}{32} \, {\left (\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}\right )} \log \left (1024 \, x^{2} + 1024 \, x \sqrt {-\sqrt {2} + 2} + 1024\right ) - \frac {1}{32} \, {\left (\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}\right )} \log \left (1024 \, x^{2} - 1024 \, x \sqrt {-\sqrt {2} + 2} + 1024\right ) + \frac {1}{32} \, \sqrt {2} \sqrt {-\sqrt {2} + 2} \log \left (256 \, x^{2} + 128 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 128 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 256\right ) + \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} + 2} \log \left (256 \, x^{2} + 128 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 128 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 256\right ) - \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} + 2} \log \left (256 \, x^{2} - 128 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 128 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 256\right ) - \frac {1}{32} \, \sqrt {2} \sqrt {-\sqrt {2} + 2} \log \left (256 \, x^{2} - 128 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 128 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 19, normalized size = 0.05 \begin {gather*} \operatorname {RootSum} {\left (1048576 t^{8} + 1, \left ( t \mapsto t \log {\left (4096 t^{5} + 4 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.01, size = 247, normalized size = 0.71 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.28, size = 311, normalized size = 0.90 \begin {gather*} -\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\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{2}+\frac {x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{2}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{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 (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 (-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 (\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 (\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 (-\frac {1}{2}-\mathrm {i}\right )+\sqrt {2}\,{\left (\sqrt {2}+2\right )}^{3/2}\,\left (\frac {1}{4}+\frac {3}{4}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}-2}}{16}+\frac {\sqrt {\sqrt {2}+2}}{16}\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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