Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{73}-7}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right ),\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]
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Rubi [A] time = 0.064989, antiderivative size = 45, 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} \[ \sqrt{\frac{2}{\sqrt{73}-7}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
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{7+\sqrt{73}-4 x^2} \sqrt{-7+\sqrt{73}+4 x^2}} \, dx\\ &=\sqrt{\frac{2}{-7+\sqrt{73}}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right )\\ \end{align*}
Mathematica [C] time = 0.0442848, size = 52, normalized size = 1.16 \[ -i \sqrt{\frac{2}{7+\sqrt{73}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{\sqrt{73}-7}}\right ),\frac{1}{12} \left (7 \sqrt{73}-61\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.255, size = 84, normalized size = 1.9 \begin{align*} 6\,{\frac{\sqrt{1- \left ( -7/6+1/6\,\sqrt{73} \right ){x}^{2}}\sqrt{1- \left ( -1/6\,\sqrt{73}-7/6 \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-42+6\,\sqrt{73}},{\frac{7\,i}{12}}\sqrt{6}+i/12\sqrt{438} \right ) }{\sqrt{-42+6\,\sqrt{73}}\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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