Optimal. Leaf size=100 \[ \frac {x^2}{2}+\frac {\tan ^{-1}\left (1-\sqrt {2} x^2\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} x^2\right )}{4 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {281, 327, 217,
1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\sqrt {2} x^2\right )}{4 \sqrt {2}}-\frac {\text {ArcTan}\left (\sqrt {2} x^2+1\right )}{4 \sqrt {2}}+\frac {x^2}{2}+\frac {\log \left (x^4-\sqrt {2} x^2+1\right )}{8 \sqrt {2}}-\frac {\log \left (x^4+\sqrt {2} x^2+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 281
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^9}{1+x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{1+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{4} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,x^2\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,x^2\right )-\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {2}}\\ &=\frac {x^2}{2}+\frac {\log \left (1-\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x^2\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x^2\right )}{4 \sqrt {2}}\\ &=\frac {x^2}{2}+\frac {\tan ^{-1}\left (1-\sqrt {2} x^2\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} x^2\right )}{4 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x^2+x^4\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 191, normalized size = 1.91 \begin {gather*} \frac {1}{16} \left (8 x^2+2 \sqrt {2} \tan ^{-1}\left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right )+2 \sqrt {2} \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-x \csc \left (\frac {\pi }{8}\right )\right )+2 \sqrt {2} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right )-2 \sqrt {2} \tan ^{-1}\left (x \sec \left (\frac {\pi }{8}\right )-\tan \left (\frac {\pi }{8}\right )\right )+\sqrt {2} \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right )+\sqrt {2} \log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt {2} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right )-\sqrt {2} \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 66, normalized size = 0.66
method | result | size |
risch | \(\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} \right )\right )}{8}\) | \(28\) |
default | \(\frac {x^{2}}{2}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x^{4}+x^{2} \sqrt {2}}{1+x^{4}-x^{2} \sqrt {2}}\right )+2 \arctan \left (1+x^{2} \sqrt {2}\right )+2 \arctan \left (-1+x^{2} \sqrt {2}\right )\right )}{16}\) | \(66\) |
meijerg | \(\frac {x^{2}}{2}-\frac {x^{2} \left (-\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}+\sqrt {x^{8}}\right )}{2 \left (x^{8}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}}\right )}{\left (x^{8}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}+\sqrt {x^{8}}\right )}{2 \left (x^{8}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{8}\right )^{\frac {1}{4}}}\right )}{\left (x^{8}\right )^{\frac {1}{4}}}\right )}{8}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 85, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{4} + \sqrt {2} x^{2} + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{4} - \sqrt {2} x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 117, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{4} \, \sqrt {2} \arctan \left (-\sqrt {2} x^{2} + \sqrt {2} \sqrt {x^{4} + \sqrt {2} x^{2} + 1} - 1\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\sqrt {2} x^{2} + \sqrt {2} \sqrt {x^{4} - \sqrt {2} x^{2} + 1} + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {2} x^{2} + 4\right ) + \frac {1}{16} \, \sqrt {2} \log \left (4 \, x^{4} - 4 \, \sqrt {2} x^{2} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 85, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2} + \frac {\sqrt {2} \log {\left (x^{4} - \sqrt {2} x^{2} + 1 \right )}}{16} - \frac {\sqrt {2} \log {\left (x^{4} + \sqrt {2} x^{2} + 1 \right )}}{16} - \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x^{2} - 1 \right )}}{8} - \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x^{2} + 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 85, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{4} + \sqrt {2} x^{2} + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{4} - \sqrt {2} x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 42, normalized size = 0.42 \begin {gather*} \frac {x^2}{2}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^2\,\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^2\,\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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