Optimal. Leaf size=44 \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (\sqrt{15}-3\right )} x\right ),-4-\sqrt{15}\right ) \]
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Rubi [A] time = 0.132986, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right ) \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+6 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{6+2 \sqrt{15}-4 x^2} \sqrt{-6+2 \sqrt{15}+4 x^2}} \, dx\\ &=\sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right )\\ \end{align*}
Mathematica [C] time = 0.0520907, size = 43, normalized size = 0.98 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{3}}} x\right ),\sqrt{15}-4\right )}{\sqrt{3+\sqrt{15}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.24, size = 84, normalized size = 1.9 \begin{align*} 3\,{\frac{\sqrt{1- \left ( -1+1/3\,\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( -1-1/3\,\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-9+3\,\sqrt{15}},i/2\sqrt{6}+i/2\sqrt{10} \right ) }{\sqrt{-9+3\,\sqrt{15}}\sqrt{-2\,{x}^{4}+6\,{x}^{2}+3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}{2 \, x^{4} - 6 \, x^{2} - 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{4} + 6 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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