Optimal. Leaf size=340 \[ x+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {327, 219,
1183, 648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 219
Rule 327
Rule 632
Rule 642
Rule 648
Rule 1183
Rubi steps
\begin {align*} \int \frac {x^8}{1+x^8} \, dx &=x-\int \frac {1}{1+x^8} \, dx\\ &=x-\frac {\int \frac {\sqrt {2}-x^2}{1-\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2}+x^2}{1+\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}\\ &=x-\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\left (-1+\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}+\left (-1+\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ &=x-\frac {1}{8} \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx+\frac {1}{16} \sqrt {2-\sqrt {2}} \int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx-\frac {1}{16} \sqrt {2-\sqrt {2}} \int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx+\frac {1}{16} \sqrt {2+\sqrt {2}} \int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx-\frac {1}{16} \sqrt {2+\sqrt {2}} \int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx\\ &=x+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {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 (3-2 \sqrt {2}\right )} \text {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 (3+2 \sqrt {2}\right )} \text {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 (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )\\ &=x+\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 210, normalized size = 0.62 \begin {gather*} x-\frac {1}{4} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x-\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{4} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right ) \cos \left (\frac {\pi }{8}\right )+\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right )-\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right )-\frac {1}{4} \tan ^{-1}\left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{4} \tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{8} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{8} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\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 = 0.17, size = 24, normalized size = 0.07
method | result | size |
default | \(x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8}\) | \(24\) |
risch | \(x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8}\) | \(24\) |
meijerg | \(x -\frac {x \left (-\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}-\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}+\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {1}{8}}}\right )}{8}\) | \(271\) |
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. 1044 vs.
\(2 (246) = 492\).
time = 0.40, size = 1044, normalized size = 3.07 \begin {gather*} \frac {1}{16} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} + \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (-\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} - \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{16} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} + \sqrt {\sqrt {2} + 2} + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{16} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \arctan \left (\frac {2 \, \sqrt {2} x - \sqrt {2} \sqrt {4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4} - \sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2} - \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{64} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) - \frac {1}{64} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} + 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) + \frac {1}{64} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} - \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) + \frac {1}{64} \, {\left (\sqrt {2} \sqrt {\sqrt {2} + 2} + \sqrt {2} \sqrt {-\sqrt {2} + 2}\right )} \log \left (4 \, x^{2} - 2 \, \sqrt {2} x \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {2} x \sqrt {-\sqrt {2} + 2} + 4\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \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 {\sqrt {2} + 2} \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 {-\sqrt {2} + 2} \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 {-\sqrt {2} + 2} \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}{32} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) + \frac {1}{32} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 15, normalized size = 0.04 \begin {gather*} x + \operatorname {RootSum} {\left (16777216 t^{8} + 1, \left ( t \mapsto t \log {\left (- 8 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 240, normalized size = 0.71 \begin {gather*} -\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 289, normalized size = 0.85 \begin {gather*} x-\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}+\sqrt {2}}\right )\,\left (\frac {\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )+\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2}+\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{\sqrt {2}+\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\frac {\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )-\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{16}-\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}+\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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