Optimal. Leaf size=150 \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+3}{\left (1-\sqrt{7}\right ) x^2+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3}}\right ),\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+3}} \sqrt{2 x^4-2 x^2-3}} \]
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Rubi [A] time = 0.0212946, antiderivative size = 150, 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} \[ \frac{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3} \sqrt{\frac{\left (1+\sqrt{7}\right ) x^2+3}{\left (1-\sqrt{7}\right ) x^2+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-\left (1-\sqrt{7}\right ) x^2-3}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{\left (1-\sqrt{7}\right ) x^2+3}} \sqrt{2 x^4-2 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-2 x^2+2 x^4}} \, dx &=\frac{\sqrt{-3-\left (1-\sqrt{7}\right ) x^2} \sqrt{\frac{3+\left (1+\sqrt{7}\right ) x^2}{3+\left (1-\sqrt{7}\right ) x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-3-\left (1-\sqrt{7}\right ) x^2}}\right )|\frac{1}{14} \left (7-\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{3+\left (1-\sqrt{7}\right ) x^2}} \sqrt{-3-2 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.045838, size = 81, normalized size = 0.54 \[ -\frac{i \sqrt{-2 x^4+2 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-1}} x\right ),\frac{1}{3} \left (\sqrt{7}-4\right )\right )}{\sqrt{1+\sqrt{7}} \sqrt{2 x^4-2 x^2-3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.185, size = 84, normalized size = 0.6 \begin{align*} 3\,{\frac{\sqrt{1- \left ( -1/3-1/3\,\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -1/3+1/3\,\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{-3-3\,\sqrt{7}}x,i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{-3-3\,\sqrt{7}}\sqrt{2\,{x}^{4}-2\,{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} - 2 \, 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} - 2 \, 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} - 2 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} - 2 \, x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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