Optimal. Leaf size=356 \[ x+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]
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Rubi [A]
time = 0.23, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1381, 1435,
1183, 648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1381
Rule 1435
Rubi steps
\begin {align*} \int \frac {x^8}{1-x^4+x^8} \, dx &=x-\int \frac {1-x^4}{1-x^4+x^8} \, dx\\ &=x+\frac {\int \frac {\sqrt {3}+2 x^2}{-1-\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3}-2 x^2}{-1+\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}\\ &=x-\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (-2+\sqrt {3}\right ) x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}+\left (-2+\sqrt {3}\right ) x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (2+\sqrt {3}\right ) x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}+\left (2+\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=x+\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx-\frac {\left (-2+\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx-\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx\\ &=x-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )\\ &=x+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 59, normalized size = 0.17 \begin {gather*} x+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\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 = 0.03, size = 44, normalized size = 0.12
method | result | size |
default | \(x +\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
risch | \(x +\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
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. 720 vs.
\(2 (264) = 528\).
time = 0.36, size = 720, normalized size = 2.02 \begin {gather*} -\frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (144 \, x^{2} + 24 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 144\right ) + \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (144 \, x^{2} - 24 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 144\right ) - \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (144 \, x^{2} + 12 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144\right ) + \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (144 \, x^{2} - 12 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 144\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {12 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (2 \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} - 2\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {12 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (2 \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} + 2\right ) + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 6} {\left (2 \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} + \sqrt {3} - 2\right ) + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 6} {\left (2 \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} - \sqrt {3} + 2\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.59, size = 26, normalized size = 0.07 \begin {gather*} x + \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (9216 t^{5} - 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 = 2.89, size = 254, normalized size = 0.71 \begin {gather*} -\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 209, normalized size = 0.59 \begin {gather*} x+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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