Optimal. Leaf size=347 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
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Rubi [A]
time = 0.13, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1387, 1141,
1175, 632, 210, 1178, 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 {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 1141
Rule 1175
Rule 1178
Rule 1387
Rubi steps
\begin {align*} \int \frac {x^4}{1-x^4+x^8} \, dx &=\frac {\int \frac {x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {\int \frac {1-x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}+\frac {\int \frac {1+x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}+\frac {\int \frac {1-x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}-\frac {\int \frac {1+x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}\\ &=-\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\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 {\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 {\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 {\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 {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}\\ &=\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 {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\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 = 39, normalized size = 0.11 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\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 = 0.02, size = 40, normalized size = 0.12
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(40\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(40\) |
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. 581 vs.
\(2 (271) = 542\).
time = 0.38, size = 581, normalized size = 1.67 \begin {gather*} \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36\right ) - \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (-6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36\right ) + \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36\right ) - \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36} \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} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-6 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 36 \, x^{2} + 36} \sqrt {\sqrt {3} + 2} + \sqrt {3} + 2\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36} \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}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-3 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 36 \, x^{2} + 36} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.53, size = 24, normalized size = 0.07 \begin {gather*} \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 18432 t^{5} + 4 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.79, size = 253, normalized size = 0.73 \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 474, normalized size = 1.37 \begin {gather*} -\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\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^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\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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