Optimal. Leaf size=73 \[ -\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1093, 207, 203} \[ -\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1093
Rubi steps
\begin {align*} \int \frac {1}{-1+3 x^2+x^4} \, dx &=\frac {\int \frac {1}{\frac {3}{2}-\frac {\sqrt {13}}{2}+x^2} \, dx}{\sqrt {13}}-\frac {\int \frac {1}{\frac {3}{2}+\frac {\sqrt {13}}{2}+x^2} \, dx}{\sqrt {13}}\\ &=-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.93 \[ -\frac {\sqrt {\sqrt {13}-3} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )+\sqrt {3+\sqrt {13}} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right )}{\sqrt {26}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 132, normalized size = 1.81 \[ \frac {1}{13} \, \sqrt {26} \sqrt {\sqrt {13} - 3} \arctan \left (\frac {1}{52} \, \sqrt {26} \sqrt {13} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {13} + 3} \sqrt {\sqrt {13} - 3} - \frac {1}{26} \, \sqrt {26} \sqrt {13} x \sqrt {\sqrt {13} - 3}\right ) + \frac {1}{52} \, \sqrt {26} \sqrt {\sqrt {13} + 3} \log \left (\sqrt {26} {\left (3 \, \sqrt {13} - 13\right )} \sqrt {\sqrt {13} + 3} + 52 \, x\right ) - \frac {1}{52} \, \sqrt {26} \sqrt {\sqrt {13} + 3} \log \left (-\sqrt {26} {\left (3 \, \sqrt {13} - 13\right )} \sqrt {\sqrt {13} + 3} + 52 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 74, normalized size = 1.01 \[ -\frac {1}{26} \, \sqrt {26 \, \sqrt {13} - 78} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {13} + \frac {3}{2}}}\right ) - \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) + \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 56, normalized size = 0.77 \[ -\frac {2 \sqrt {13}\, \arctanh \left (\frac {2 x}{\sqrt {-6+2 \sqrt {13}}}\right )}{13 \sqrt {-6+2 \sqrt {13}}}-\frac {2 \sqrt {13}\, \arctan \left (\frac {2 x}{\sqrt {6+2 \sqrt {13}}}\right )}{13 \sqrt {6+2 \sqrt {13}}} \]
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 \[ \int \frac {1}{x^{4} + 3 \, x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 93, normalized size = 1.27 \[ -\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {\sqrt {13}+3}}+\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {\sqrt {13}+3}}\right )\,\sqrt {\sqrt {13}+3}}{26}-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {3-\sqrt {13}}}-\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {3-\sqrt {13}}}\right )\,\sqrt {3-\sqrt {13}}}{26} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.45, size = 146, normalized size = 2.00 \[ \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} + 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} \right )} - \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} + 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \right )} - 2 \sqrt {- \frac {3}{104} + \frac {\sqrt {13}}{104}} \operatorname {atan}{\left (\frac {2 \sqrt {2} x}{3 \sqrt {-3 + \sqrt {13}} + \sqrt {13} \sqrt {-3 + \sqrt {13}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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