Optimal. Leaf size=19 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right ),\frac{6}{5}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.0119673, antiderivative size = 19, 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, 420} \[ -\frac{F\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+7 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{12-4 x^2} \sqrt{-2+4 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\frac{x}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5}}\\ \end{align*}
Mathematica [B] time = 0.0226898, size = 58, normalized size = 3.05 \[ \frac{\sqrt{1-2 x^2} \sqrt{1-\frac{x^2}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),\frac{1}{6}\right )}{\sqrt{2} \sqrt{-2 x^4+7 x^2-3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 48, normalized size = 2.5 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{-3\,{x}^{2}+9}\sqrt{-2\,{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},\sqrt{6} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+7\,{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} + 7 \, 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} + 7 \, x^{2} - 3}}{2 \, x^{4} - 7 \, 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} + 7 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} + 7 \, x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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