Optimal. Leaf size=275 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}} \]
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Rubi [A]
time = 0.14, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1360, 1183,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1360
Rubi steps
\begin {align*} \int \frac {1}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3}-x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (-1+\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 (-1+\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 (1+\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 (1+\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 {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}-\frac {\int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{2 \sqrt {6 \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 )}{2 \sqrt {6 \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 )}{2 \sqrt {6 \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 )}{2 \sqrt {6 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}\\ \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 = 42, normalized size = 0.15 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}^3+2 \text {$\#$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 = 0.02, size = 30, normalized size = 0.11
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+3 \textit {\_R} x +x^{2}\right )\right )}{4}\) | \(30\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+3 \textit {\_R} x +x^{2}\right )\right )}{4}\) | \(30\) |
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 [A]
time = 0.35, size = 220, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \sqrt {2} \arctan \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{3} - x\right )} + x^{2} - \sqrt {x^{4} + \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1} {\left (\sqrt {3} \sqrt {2} x - 2\right )}}{3 \, x^{2} - 2}\right ) - \frac {1}{6} \, \sqrt {3} \sqrt {2} \arctan \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{3} - x\right )} - x^{2} - \sqrt {x^{4} - \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1} {\left (\sqrt {3} \sqrt {2} x + 2\right )}}{3 \, x^{2} - 2}\right ) + \frac {1}{24} \, \sqrt {3} \sqrt {2} \log \left (36 \, x^{4} + 36 \, \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 108 \, x^{2} + 36\right ) - \frac {1}{24} \, \sqrt {3} \sqrt {2} \log \left (36 \, x^{4} - 36 \, \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 108 \, x^{2} + 36\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 165, normalized size = 0.60 \begin {gather*} \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} - \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} - 4 x^{2} + 2 \sqrt {6} x - 3 \right )}\right )}{24} + \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} + \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} + 4 x^{2} + 2 \sqrt {6} x + 3 \right )}\right )}{24} - \frac {\sqrt {6} \log {\left (x^{4} - \sqrt {6} x^{3} + 3 x^{2} - \sqrt {6} x + 1 \right )}}{24} + \frac {\sqrt {6} \log {\left (x^{4} + \sqrt {6} x^{3} + 3 x^{2} + \sqrt {6} x + 1 \right )}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.92, size = 205, normalized size = 0.75 \begin {gather*} \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \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 = 0.04, size = 53, normalized size = 0.19 \begin {gather*} \sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}-\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}+\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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