Optimal. Leaf size=67 \[ \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1093, 207} \[ \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1093
Rubi steps
\begin {align*} \int \frac {1}{1-4 x^2+x^4} \, dx &=\frac {\int \frac {1}{-2-\sqrt {3}+x^2} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{-2+\sqrt {3}+x^2} \, dx}{2 \sqrt {3}}\\ &=\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 123, normalized size = 1.84 \[ -\frac {1}{12} \, \sqrt {3} \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + x\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {\sqrt {3} + 2} \log \left (-\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + x\right ) - \frac {1}{12} \, \sqrt {3} \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + x\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {-\sqrt {3} + 2} \log \left (-{\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 101, normalized size = 1.51 \[ \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left ({\left | x + \frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} \right |}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left ({\left | x + \frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left ({\left | x - \frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left ({\left | x - \frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.90 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}-\frac {\sqrt {3}\, \arctanh \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \left (\sqrt {6}+\sqrt {2}\right )} \]
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 \[ \int \frac {1}{x^{4} - 4 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 98, normalized size = 1.46 \[ \mathrm {atanh}\left (\frac {5\,\sqrt {2}\,x}{\sqrt {2}\,\sqrt {6}+4}+\frac {3\,\sqrt {6}\,x}{\sqrt {2}\,\sqrt {6}+4}\right )\,\left (\frac {\sqrt {2}}{4}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,x}{\sqrt {2}\,\sqrt {6}-4}-\frac {3\,\sqrt {6}\,x}{\sqrt {2}\,\sqrt {6}-4}\right )\,\left (\frac {\sqrt {2}}{4}-\frac {\sqrt {6}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 24, normalized size = 0.36 \[ \operatorname {RootSum} {\left (2304 t^{4} - 192 t^{2} + 1, \left (t \mapsto t \log {\left (384 t^{3} - 28 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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