Optimal. Leaf size=180 \[ \frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1437, 1175,
632, 210, 1178, 642} \begin {gather*} \frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {3 \left (2-\sqrt {3}\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 1175
Rule 1178
Rule 1437
Rubi steps
\begin {align*} \int \frac {3-2 \sqrt {3}+\left (-3+\sqrt {3}\right ) x^4}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3} \left (3-2 \sqrt {3}\right )+\left (-6+3 \sqrt {3}\right ) x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3} \left (3-2 \sqrt {3}\right )+\left (6-3 \sqrt {3}\right ) x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {1}{4} \sqrt {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}{4} \sqrt {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}{4} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {1}{4} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx\\ &=\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{2} \left (3-2 \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}{2} \left (3-2 \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}{2} \sqrt {6-3 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \sqrt {6-3 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {3 \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{4} \sqrt {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 = 0.03, size = 89, normalized size = 0.49 \begin {gather*} \frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})-2 \sqrt {3} \log (x-\text {$\#$1})-3 \log (x-\text {$\#$1}) \text {$\#$1}^4+\sqrt {3} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$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 = 0.09, size = 62, normalized size = 0.34
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-6 \textit {\_R}^{4}+2 \sqrt {3}\, \textit {\_R}^{4}+\left (-3+\sqrt {3}\right ) \left (\sqrt {3}-1\right )\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{8}\) | \(62\) |
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 = 0.37, size = 181, normalized size = 1.01 \begin {gather*} -\frac {1}{2} \, \sqrt {-3 \, \sqrt {3} + 6} \arctan \left (\frac {1}{3} \, {\left (3 \, x^{3} + \sqrt {3} {\left (2 \, x^{3} - x\right )} - 3 \, x\right )} \sqrt {-3 \, \sqrt {3} + 6}\right ) - \frac {1}{2} \, \sqrt {-3 \, \sqrt {3} + 6} \arctan \left (\frac {1}{3} \, {\left (2 \, \sqrt {3} x + 3 \, x\right )} \sqrt {-3 \, \sqrt {3} + 6}\right ) + \frac {1}{4} \, \sqrt {-3 \, \sqrt {3} + 6} \log \left (\frac {3 \, x^{8} + 12 \, x^{6} + 15 \, x^{4} + 12 \, x^{2} - 6 \, \sqrt {3} {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} + 2 \, {\left (3 \, x^{5} + 3 \, x^{3} - \sqrt {3} {\left (x^{7} + x^{5} + x^{3} + x\right )}\right )} \sqrt {-3 \, \sqrt {3} + 6} + 3}{x^{8} - x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.95, size = 131, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\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}{8} \, {\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 = 2.23, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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