Optimal. Leaf size=346 \[ -\frac {1}{5 x^5}-\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )-\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
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Rubi [A]
time = 0.12, antiderivative size = 346, 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 = {331, 305,
1108, 648, 632, 210, 642} \begin {gather*} -\frac {1}{8} \sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{5 x^5}-\frac {\log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 305
Rule 331
Rule 632
Rule 642
Rule 648
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (1+x^8\right )} \, dx &=-\frac {1}{5 x^5}-\int \frac {x^2}{1+x^8} \, dx\\ &=-\frac {1}{5 x^5}-\frac {\int \frac {1}{1-\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}+\frac {\int \frac {1}{1+\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}\\ &=-\frac {1}{5 x^5}+\frac {\int \frac {\sqrt {2-\sqrt {2}}-x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}-x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}+x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ &=-\frac {1}{5 x^5}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ &=-\frac {1}{5 x^5}-\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 x\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 x\right )}{4 \sqrt {2}}\\ &=-\frac {1}{5 x^5}-\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 {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 216, normalized size = 0.62 \begin {gather*} -\frac {1}{5 x^5}-\frac {1}{4} \tan ^{-1}\left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \cos \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 ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right )+\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right )+\frac {1}{4} \tan ^{-1}\left (\sec \left (\frac {\pi }{8}\right ) \left (x-\sin \left (\frac {\pi }{8}\right )\right )\right ) \sin \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 ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{8} \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{8} \log \left (1+x^2+2 x \cos \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 = 0.18, size = 28, normalized size = 0.08
method | result | size |
risch | \(-\frac {1}{5 x^{5}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3}+x \right )\right )}{8}\) | \(26\) |
default | \(-\frac {1}{5 x^{5}}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{8}\) | \(28\) |
meijerg | \(-\frac {1}{5 x^{5}}-\frac {x^{3} \left (-\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}+\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}-\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}-\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}-\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}+\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {3}{8}}}\right )}{8}\) | \(277\) |
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. 1587 vs.
\(2 (250) = 500\).
time = 0.41, size = 1587, normalized size = 4.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.16, size = 22, normalized size = 0.06 \begin {gather*} \operatorname {RootSum} {\left (16777216 t^{8} + 1, \left ( t \mapsto t \log {\left (512 t^{3} + x \right )} \right )\right )} - \frac {1}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 244, 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 ) - \frac {1}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 196, normalized size = 0.57 \begin {gather*} \mathrm {atan}\left (x\,{\left (\frac {\sqrt {-\sqrt {2}-2}}{16}-\frac {\sqrt {2-\sqrt {2}}}{16}\right )}^5\,32768{}\mathrm {i}\right )\,\left (\frac {\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )+\mathrm {atan}\left (x\,{\left (\frac {\sqrt {\sqrt {2}-2}}{16}+\frac {\sqrt {\sqrt {2}+2}}{16}\right )}^5\,32768{}\mathrm {i}\right )\,\left (\frac {\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )-\frac {1}{5\,x^5}+\mathrm {atan}\left (x\,{\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )}^5\,{\left (\sqrt {2}+2\right )}^{5/2}\,32768{}\mathrm {i}\right )\,\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}+\mathrm {atan}\left (x\,{\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{16}-\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )}^5\,{\left (\sqrt {2}+2\right )}^{5/2}\,32768{}\mathrm {i}\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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