Optimal. Leaf size=148 \[ \frac{\sqrt{\frac{3-\left (3-\sqrt{15}\right ) x^2}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{\left (3+\sqrt{15}\right ) x^2-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{15} x}{\sqrt{\left (3+\sqrt{15}\right ) x^2-3}}\right ),\frac{1}{10} \left (5+\sqrt{15}\right )\right )}{\sqrt{2} 3^{3/4} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{2 x^4+6 x^2-3}} \]
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Rubi [A] time = 0.0397224, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1098} \[ \frac{\sqrt{\frac{3-\left (3-\sqrt{15}\right ) x^2}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{\left (3+\sqrt{15}\right ) x^2-3} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{15} x}{\sqrt{\left (3+\sqrt{15}\right ) x^2-3}}\right )|\frac{1}{10} \left (5+\sqrt{15}\right )\right )}{\sqrt{2} 3^{3/4} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{2 x^4+6 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+6 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3-\left (3-\sqrt{15}\right ) x^2}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{-3+\left (3+\sqrt{15}\right ) x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{15} x}{\sqrt{-3+\left (3+\sqrt{15}\right ) x^2}}\right )|\frac{1}{10} \left (5+\sqrt{15}\right )\right )}{\sqrt{2} 3^{3/4} \sqrt [4]{5} \sqrt{\frac{1}{3-\left (3+\sqrt{15}\right ) x^2}} \sqrt{-3+6 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0645388, size = 77, normalized size = 0.52 \[ -\frac{i \sqrt{-2 x^4-6 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\sqrt{\frac{5}{3}}-1} x\right ),-4-\sqrt{15}\right )}{\sqrt{\sqrt{15}-3} \sqrt{2 x^4+6 x^2-3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.184, size = 84, normalized size = 0.6 \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\,\sqrt{9-3\,\sqrt{15}}x,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{1}{\sqrt{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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