Optimal. Leaf size=100 \[ -\frac {1}{6 x^6}+\frac {\tan ^{-1}\left (1-\sqrt {2} x^2\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} x^2+1\right )}{4 \sqrt {2}}+\frac {\log \left (x^4-\sqrt {2} x^2+1\right )}{8 \sqrt {2}}-\frac {\log \left (x^4+\sqrt {2} x^2+1\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.07, 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 = {275, 325, 211, 1165, 628, 1162, 617, 204} \[ -\frac {1}{6 x^6}+\frac {\log \left (x^4-\sqrt {2} x^2+1\right )}{8 \sqrt {2}}-\frac {\log \left (x^4+\sqrt {2} x^2+1\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} x^2\right )}{4 \sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} x^2+1\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 275
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (1+x^8\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{6 x^6}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{6 x^6}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{6 x^6}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,x^2\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,x^2\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {2}}\\ &=-\frac {1}{6 x^6}+\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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x^2\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x^2\right )}{4 \sqrt {2}}\\ &=-\frac {1}{6 x^6}+\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 = 0.08, size = 193, normalized size = 1.93 \[ \frac {1}{48} \left (-\frac {8}{x^6}-3 \sqrt {2} \log \left (x^2-2 x \sin \left (\frac {\pi }{8}\right )+1\right )-3 \sqrt {2} \log \left (x^2+2 x \sin \left (\frac {\pi }{8}\right )+1\right )+3 \sqrt {2} \log \left (x^2-2 x \cos \left (\frac {\pi }{8}\right )+1\right )+3 \sqrt {2} \log \left (x^2+2 x \cos \left (\frac {\pi }{8}\right )+1\right )-6 \sqrt {2} \tan ^{-1}\left (x \sec \left (\frac {\pi }{8}\right )-\tan \left (\frac {\pi }{8}\right )\right )+6 \sqrt {2} \tan ^{-1}\left (\csc \left (\frac {\pi }{8}\right ) \left (x+\cos \left (\frac {\pi }{8}\right )\right )\right )+6 \sqrt {2} \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-x \csc \left (\frac {\pi }{8}\right )\right )+6 \sqrt {2} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 125, normalized size = 1.25 \[ \frac {12 \, \sqrt {2} x^{6} \arctan \left (-\sqrt {2} x^{2} + \sqrt {2} \sqrt {x^{4} + \sqrt {2} x^{2} + 1} - 1\right ) + 12 \, \sqrt {2} x^{6} \arctan \left (-\sqrt {2} x^{2} + \sqrt {2} \sqrt {x^{4} - \sqrt {2} x^{2} + 1} + 1\right ) - 3 \, \sqrt {2} x^{6} \log \left (x^{4} + \sqrt {2} x^{2} + 1\right ) + 3 \, \sqrt {2} x^{6} \log \left (x^{4} - \sqrt {2} x^{2} + 1\right ) - 8}{48 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 85, normalized size = 0.85 \[ -\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{4} + \sqrt {2} x^{2} + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{4} - \sqrt {2} x^{2} + 1\right ) - \frac {1}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 71, normalized size = 0.71 \[ -\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x^{2}-1\right )}{8}-\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x^{2}+1\right )}{8}-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+\sqrt {2}\, x^{2}+1}{x^{4}-\sqrt {2}\, x^{2}+1}\right )}{16}-\frac {1}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 85, normalized size = 0.85 \[ -\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} + \sqrt {2}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x^{2} - \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{4} + \sqrt {2} x^{2} + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{4} - \sqrt {2} x^{2} + 1\right ) - \frac {1}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 42, normalized size = 0.42 \[ -\frac {1}{6\,x^6}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^2\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x^2\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 87, normalized size = 0.87 \[ \frac {\sqrt {2} \log {\left (x^{4} - \sqrt {2} x^{2} + 1 \right )}}{16} - \frac {\sqrt {2} \log {\left (x^{4} + \sqrt {2} x^{2} + 1 \right )}}{16} - \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x^{2} - 1 \right )}}{8} - \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x^{2} + 1 \right )}}{8} - \frac {1}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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